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LIBRARY  of  psychology  and  scientific  methods 

Edited  by  J.  McKEEN  CATTELL 


THEORY  OF  MENTAL  AND  SOCIAL 
MEASUREMENTS 


AN  INTRODUCTION 


TO  THE 


THEORY  OF  MENTAL  AND  SOCIAL 


MEASUREMENTS 


BY 

EDWARD   L.  THORNDIKE 

PROFESSOR  OF  EDUCATIONAL  PSYCHOLOGY   IN  TEACHERS  COLLEGE 
COLUMBIA  UNIVERSITY 


IS5  2.  / 


NEW   YORK 
THE  SCIENCE   PRESS 

1904 

1907 


Copyright,  1904 
BY  EDWARD  L.  THORNDIKE 


Press  of 

The  New  Era  Printihg  Compaut, 

Lancaster,  Pa. 


ILLUSTRATIONS. 


1-11.     Illustrations  of  distributions  of  traits  in  individuals. 

1.  Memory  span  of  B.  F.  A. 
Ia.     Memory  span  of  B.  F.  A. 

2.  Accuracy  of  discrimination  of  length  of  E.  H. 
2a.     Accuracy  of  discrimination  of  length  of  E.  H. 

3.  The  condition  of  labor  in  the  case  of  the   Amalgamated  Society  of 

Engineers. 
3A.     The  condition  of  labor  in  the  case  of  the  Amalgamated  Society  of 
Engineers. 

4.  Reaction-time  of  H. 

5.  Quickness  of  Movement  of  T. 

6.  Ability  in  addition  of  J.  S. 

7.  Efficiency  of  perception  of  S. 

8.  Condition  of  a  trade. 

9.  Attendance  of  a  school. 

10.  Receipts  of  a  sub-treasury. 

11.  Pulse  of  B. 

12.  Typical  form  of  distribution. 

13.  Discrimination  of  length  in  11  individuals. 

14.  Types  of  asymmetrical  distribution. 

15.  Two  distributions  differing  only  in  variability. 

16.  A  form  of  distribution  to  illusti-ate  the  possibility  of  simple  mathe- 

matical expression  of  a  variable  quantity. 

17.  A  form  of  distribution  to  illustrate  the  possibility  of  simple  mathe- 

matical expression  of  a  variable  quantity. 

18.  19,  &  20.     Comparison  of  distributions  based  on  few  and  on  many  rec- 

ords from  each  member  of  a  group. 

21.  Distribution  of  height  of  adult  men. 

22.  Distribution  of  weight  of  adult  men. 

23.  Distribution  of  cephalic  index  of  Alt-Bayerische  skulls. 

24.  Distribution  of  length  of  male  infants. 

25.  Distribution  of  girth  of  chest  of  adult  men. 

26.  Distribution  of  strength  of  arm  pull. 

27.  Distribution  of  body  temperature  at  the  mouth. 

28.  Distribution  of  heart  rate. 

29.  Distribution  of  reaction  time  of  college  freshmen. 

30.  Distribution  of  memory  span  for  digits. 

31.  Distribution  of  perceptive  ability  of  L2-year-old  hoys. 

32.  Distribution  of  ability  in  controlled  association. 

33.  Distribution  of  accuracy  of. discrimination  of  length. 

ix 


x  I  LUSTRATIONS. 

34.     Distribution  of  perceptive  ability. 

85.      Distribution  of  ratio  of  attendance  to  enrollment  in  city  schools. 

St;.     Distribution  of  wage  of  cotton  operatives. 

87.      Distribution  of  age  of  graduation  from  American  colleges. 

38.     Distribution  of  cost  per  pupil  of  education. 

89   40.     Distribution  of  wages  of  American  workingmen. 

41.  Distribution  of  ratio  of  attendance  to  enrollment  in  city  schools. 

42.  Distribution  of  income  of  American  colleges. 

43.  Distribution  of  age  at  marriage  of  gifted  men. 

44.  Distribution  of  period  between  marriage  and  divorce. 

45.  Distribution  of  size  of  New  England  families. 

46.  Distribution  of  infant  mortality. 

47.  Distribution  of  age  at  death. 

48.  An  irregular  distribution.     (Hypothetical.) 

49.  A  regular  distribution.     (Hypothetical.) 

50.  The  influence  of  combination  of  species  upon  the  form  of  distribution 

of  a  group. 

51  52  53  and  54.  The  influence  of  selection  upon  the  form  of  distribu- 
tion of  a  group. 

55-67.  The  influence  upon  the  form  of  distribution  of  the  number  and 
nature  of  causes  producing  the  quantity. 

68.  Form  of  distribution  assumed  as  an  illustrative  case  of  the  possibility 

of  transmuting  any  ordered  series  of  known  form  of  distribution. 

69.  Fig.  68  approximated  by  a  series  of  rectangles. 

70.  Fig.  68  with  base  line  divided  to  correspond  to  Table  XXIX. 
71-76.     Illustrations  of  the  inadecpuacy  of  a  comparison  of  groups  by  their 

averages  alone. 

77.  Comparison  of  boys  and  girls  with  respect  to  the  A  test. 

78.  Change  in  average  height  of  25  boys  during  5  years. 

79.  Change  in  each  of  25  boys  in  height  during  5  years. 

80.  Change  in  each  of  25  boys  in  height  from  12  to  14  and  from  14  to  16. 

81.  The  relation  between  the  refraction  of  air  and  its  density. 

82.  The  relation  between  lapse  of  time  and  memory. 

83.  The  relation  between  two  perceptive  abilities. 

84.  The  same  facts  as  in  Fig.  83,  but  referred  to  central  axes. 

85.  The  facts  of  Figs.  83  and  84  with  the  individual  measures  of  each 

array  replaced  by  their  central  tendency. 

86.  Comparison  of  two  groups. 

87.  Forms  of  distribution. 


PREFACE. 


Experience  has  sufficiently  shown  that  the  facts  of  human  nature 
can  be  made  the  material  for  quantitative  science.  The  direct  trans- 
fer of  methods  originating  in  the  physical  sciences  or  in  commercial 
arithmetic  to  sciences  dealing  with  the  complex  and  variable  facts  of 
human  life  has,  however,  resulted  in  crude  and  often  fallacious  meas- 
urements. Moreover,  it  has  been  difficult  to  teach  students  to  esti- 
mate quantitative  evidence  properly  or  to  obtain  and  use  it  wisely, 
because  the  books  to  which  one  could  refer  them  were  too  abstract 
mathematically  or  too  specialized,  and  omitted  altogether  much  of  the 
knowledge  about  mental  measurements  most  needed  by  the  majority 
of  university  students. 

It  is  the  aim  of  this  book  to  introduce  students  to  the  theory  of 
mental  measurements  and  to  provide  them  with  such  knowledge  and 
practice  as  may  assist  them  to  follow  critically  quantitative  evidence 
and  argument  and  to  make  their  own  researches  exact  and  logical. 
Only  the  most  general  principles  are  outlined,  the  special  methods 
appropriate  to  each  of  the  mental  sciences  being  better  left  for  sepa- 
rate treatment.  If  the  general  problems  of  mental  measurement  are 
realized  and  the  methods  at  hand  for  dealing  with  variable  quantities 
are  mastered,  the  student  will  find  no  difficulty  in  acquiring  the 
special  information  and  technique  involved  in  the  quantitative  aspect 
of  his  special  science.  The  author  has  had  in  mind  the  needs  of  stu- 
dents of  economics,  sociology  and  education,  possibly  even  nunc  than 
those  of  students  of  {psychology,  pure  and  simple.  Indeed,  a  great 
part  of  the  discussion  is  relevant  to  the  problems  of*  anthropometry 
and  vital  statistics.  The  book  may  with  certain  limitations  be  u^vd 
as  an  introduction  to  the  theory  of  measuremenl  of  all  variable  phe- 
nomena. 


CONTENTS. 


Chapter.  Page. 

I.     Introduction 1 

II.  Units  of  Measurement.     ........       7 

III.  The  Measurement  of  an  Individual.         .         .         .         .         .22 

IV.  The  Measurement  of  a  Group 41 

V.  The  Causes  of  Variability  and  the  Application  of  the  Theory  of 

Probability  to  Mental  Measurements 61 

VI.     The  Arithmetic_of  Calculating  Central  Tendencies  and  Varia- 
bilities. ..........     71 

VII.     The   Transmutation   of    Measures  by  Relative   Position   into 

Terms  of  Units  of  Amount. 85 

VIII.     The  Measurement  of  Differences  and  of  Changes.     .         .         .97 
IX.     The  Measurement  of  Relationships.         .         .         .         .         .110 

X.     The   Reliability   of    Measures 136 

XL     The  Use  of  Tables  of  Frequency  of  the  Probability  Surface.      .   147 
XII.     Sources  of  Error  in  Measurements.  ......   157 

XIII.     Conclusion.     References  for  further  study.       ....  163 

Appendix. 

I.     A  Multiplication  Table  up  to  100  X  100 169 

II.     A  Table  of  Squares  and  Square  Roots  up  to  1,000.  .         .  I9j0 

III.     Answers  to   Problems.     Miscellaneous  Problems.  .         .  201 


VI 1 


TABLES. 


I.     Variation  in  units  of  spelling  ability. 
II.     Variation  in  opinions  concerning  units  of  arithmetical  ability. 

III.  Variation  in  opinions  concerning  units  of  ability  in  controlled 

association  of  ideas. 

IV.  Variation  in  opinions  concerning  units  of  ability  in  controlled 

association  of  ideas. 
V.     Variation  in  opinions  concerning  units  of  ability  in  controlled 

association  of  ideas. 
VI.     Memory  span  of  B.  F.  A. 
VII.     Accuracy  of  discrimination  of  length  of  E.  H. 
VIII.     The  condition  of  labor  in  the  case  of  the  Amalgamated  Society 
of  Engineers. 
IX.     Eeaction-time  of  H. 
X.     Quickness  of  movement  of  T. 
XI.     Ability  in  addition  of  J.  S. 
XII.     Efficiency  of  perception  of  S. 

XIII.  The  condition  of  labor  in  the  case  of  the  Friendly  Society  of 

Iron  Founders. 

XIV.  Attendance  of  a  school. 

XV.     Sub-treasury  daily  receipts  from  banks. 
XVI.     Pulse  of  B. 

XVII.     Discrimination  of  length  in  11  individuals. 
XVIII.     Two  abilities  differing  only  in  variability. 
XIX.     Illustration  of  the  trustworthiness  of  an  average  of  a  group, 

calculated  from  a  few  records  for  each  member. 
XX.     Illustration  of  the  trustworthiness  of  an  average  of  a  -roup, 
calculated  from  a  few  records  for  each  member. 
XXI.     Illustration  of  the  trustworthiness  of  a  distribution,  calculated 

from  only  a  few  records  for  each  member. 
XXII.     Table  of  frequencies  of  the  normal   probability   surface   in 
terms  of  A.  D. 

XXIII.  Table  of  frequencies  of  the  normal  probability  surface  in 

terms  of  a. 

XXIV.  Combinations  of  4  cau  i    , 
XXV.     Combinations  of  5  causes. 

XXVI.     Combinations  of  6  causes. 
XXVII.     Combinations  «»f  1".  L5  and  20  causes. 
XXVIII.     Cost  of  education   per  pupil   for  lull  year'     attendance  in 

American   cities. 
XXIX.      100  boys  ranked  by  their  serial  urder  with  re-]..,  |   lu  a  Hail. 

xi 


Xll 


TABLES. 


XXX.     Relative  frequencies  of  equidistant  abilities  of  the  surface  of 
frequency  of  Fig.  69. 
XXXI.     Average  distance  from  the  average  of  each  serial  per  cent,  of 

the  cases  in  any  normal  distribution. 
XXXII.     Average  distance  from  the  average  of  any  continuous  group 
in  a  normal  distribution. 

XXXIII.  Abilities  in  the  A  test  of  boys  and  girls. 

XXXIV.  Growth  of  25  boys  from  the  13th  through  the  17th  year. 
XXXV.     The  relation  between  the  lapse  of  time  and  memory. 

XXXVI.     The  relation  between  two  perceptive  abilities. 
XXXVII.     The  relation  between  two  perceptive  abilities,  each  being  ex- 
pressed in  terms  of  deviations  from  the  central  tendency  as 
a  zero  point. 
XXXVIII.     The  relationship  of  Tables  XXXVI.  and  XXXVII.  expressed 
in  a  series  of  measures  each  with  the  average  of  its  related 
array. 
XXXIX.     The  relationship  of  Tables  XXXVI.  and  XXXVII.  expressed 
as  a  series  of  individual  ratios. 
XL.     Table  XXXIX. ,  with  allowance  made  for  the  variability  of 

each  trait. 
XLI.     Calculation  of  r,  using  averages  of  arrays. 
XLII.     Calculation  of  r,  using  individual  records. 
XLIII.     Table  of  values  of  the  normal  probability  integral,  correspond- 
ing to  values  of  x\a. 
XLIV.     Table  of  values  of  the  normal  probability  integral,  correspond- 
ing to  values  of  x\K.  D. 
XLV.     Table  of  values  of  the  normal  probability  integral,  correspond- 
ing to  values  of  z/P.  E. 
XLVI.     Table  of  frequencies  of  normal  surface  with  Av.  =  24,  and 
<r  =  4. 
XLVII.     Comparison  of  an  actual  distribution  with  the  normal  distri- 
bution of  the  same  Av.  and  A.  D. 


CHAPTER  I. 

INTRODUCTION. 

/  6S-2. 1 
Mathematics  and  Measurements. 

The  power  to  follow  abstract  mathematical  arguments  is  rare 
and  its  development  in  the  course  of  school  education  is  rarer  still. 
For  example,  few  of  us  are  able  to  understand  the  symbols  or  pro- 
cesses used  in  the  quotation  on  the  following  page.  Yet  it  is  a  rather 
easy  sample  of  the  discussions  from  which  the  student  is  expected  to 
gain  insight  into  the  theory  of  measurement  appropriate  to  the  vari- 
able phenomena  with  which  the  mental  sciences  have  to  deal. 

It  would  be  unfortunate  if  the  ability  to  understand  and  use  the 
newer  methods  of  measurement  were  dependent  upon  the  mathemat- 
ical capacity  and  training  which  were  required  to  derive  and  formu- 
late them.  The  great  majority  of  thinkers  would  then  be  deprived 
of  the  most  efficient  weapon  in  investigations  of  mental  and  social 
facts,  and  adequate  statistical  studies  could  be  made  only  by  the  few 
students  of  psychology,  sociology,  economics  and  education  who  hap- 
pened to  be  also  proficient  mathematicians. 

There  is,  happily,  nothing  in  the  general  principles  of  modern 
statistical  theory  but  refined  common  sense,  and  little  in  the  tech- 
nique resulting  from  them  that  general  intelligence  can  not  readily 
*«.  master.     A  new  method  devised  by  a  mathematician  is  likely  to  be 
>  expressed  by  him  in  terms  intelligible  only  to  those  with  mathemat- 
v)  ical  training,  and  to  be  explained  by  him  through  an  abstract  deri- 
vation which  only  those  with  mathematical  training  and  capacity 
can  understand.     It  may,  nevertheless,  be  possible  to   explain   its 
meaning  and  use  in  common  language  to  a  common-sense  thinker. 
With  time  what  were  the  mysteries  of  the  specialist  become  the  prop- 
erty of  all.     To  aid  this  process  in  the  case  of  certain  recent  contri- 
butions to  statistical  theory  is  one  of  the  leading  aims  of  this  book. 
Knowledge  will  be  presupposed  of  only  the  elements  of  arithmetic 
and  algebra.     Artificial   symbols  will   be  used  only  when   they  arc 
really   convenient.      Concrete   illustrations   will   always   accompany 
and  often  replace  abstract  laws. 
1  1 


MENTAL    AND    SOCIAL    M I.Wsci; i:.Mh\xrs 

Deduction  of  Equation  of  Curve  of  Error,  from  A.  L.  Bowley's 
•  Elements  of  Statistics,5  p.  275  f. 

We  ran  now  proceed  to  the  determination  of  the  equation  of  the 
curve  of  error. 

'The  chance  oi'  r  successes  is  greatest  when  r  is  the  greatest  integer 
in  /at  ;  this  is  found  by  the  ordinary  method  of  determining  the 
maximum  term  in  a  binomial  expansion. 

Let   P  be  this  maximum  value  =  "( ■    ■  j>'"'  </'",  taking  the  supposi- 
tion tor  brevity  that  ////  is  integral,  which  will  not  affect  the  proof. 
n 
=    -'~- —  ppn<iq",  for  on  -f  an  =  n. 
/pn  jqn1     J  J 

Lei  /'  be  chance  of  pn  4-  sc  white  balls.     Then 

(p\*        qn(qn-l)---(qn-x-\-  1) 
"  X\q)    X(Pn  +  l)(pn  +  2)...(pn  +  x) 

=Px       V    ~9»A     ~Qn)'"\    ~     an    ) 

Taking  logarithms  of  both  sides 
logPx  =  logP+log(l-^)  +  log(l-J)  +  -.. 

+lo^1-a^)-log(1+i)-log(1+F0 

_..._log(1  +  ^l)_log(l+^)      ^ 

~  \pn  ~~  2    '  (pnjz  +  )  ~  \pn  ~  2  \pn)       J 

~  \pn      2  \pn  J       J 

1  +  2+ •••  +  (*-!)      l2+22+---  +  fo-l)2 

1  +  2  +  •    ■  +x      l  +  22  +  ---  +  «:2 

+  0~A„2 


=  logP 


pn  T  2pW 

a;  (as  —  1)      cc  (re  +  1)       («  —  !)•«?•  (2a;  —  1) 


2^2 


2  7//  2^>w  12§2n: 


a;(a;-H)(2a;  +  l) 

12/n2  + 


INTRODUCTION  3 

Let  no  one  suppose  that  the  foregoing  statements  imply  that 
mathematical  gifts  and  training  are  useless  possessions  for  a  student 
of  quantitative  mental  science.  On  the  contrary,  the  assumption  of 
their  absence  in  '  the  reader '  will  necessitate  long  descriptions,  round- 
about arguments  and  awkward  formula?.  If  this  book  were  written 
by  a  mathematician  for  the  mathematically  minded  it  would  not 
need  to  be  one  fifth  as  long.  If  it  is  read  by  such  a  one,  it  may 
well  seem  intolerably  clumsy  and  inelegant. 

General  Information  about  Measwrements, 

There  are,  in  addition  to  the  recent  studies  of  the  general  theory 
of  mental  measurements,  a  number  of  matters  concerning  the  quanti- 
tative treatment  of  human  nature  which  sufficient  experience  teaches 
thoughtful  workers  everywhere,  but  which  have  not  been  stated 
simply  and  conveniently  in  available  form  for  study  and  reference. 
At  present  one  must  learn  these  gradually  and  with  difficulty  by 
himself  or  acquire  them  from  the  oral  traditions  of  the  laboratory  or 
class-room.  They  are,  for  the  most  part,  extremely  simple.  But 
that  one  sees  them  at  the  first  glance  when  they  are  presented  does 
not  imply  that  he  would  not  in  nine  cases  out  of  ten  fail  to  discover 
them  if  they  were  not  presented.  To  put  these  at  the  service  of  all 
who  need  to  know  about  them  is  the  second  aim  of  this  book. 

The  Technique  of  Measurements. 
Although  the  formula?  used  in  expressing  and  comparing  mental 
measurements  are  in  most  cases  straightforward  and  simple,  they  are 
often  so  foreign  to  the  habits  acquired  in  connection  with  the  arith- 
metic and  algebra  of  one's  school  days  that  ready  and  sure  use  of 
them  can  be  acquired  only  by  practice.  Convenient  and  accurate 
manipulation  of  figures  is  one  of  the  many  things  which  one  learns 
to  do  by  doing.  A  mere  statement  of  a  rule  leaves  one  uncertain. 
Only  after  applying  it  a  number  of  times  does  he  really  possess  it. 
For  example,  I  doubt  if  any  one  of  my  readers  is  sure  that  from  a 
mere  reading  he  understands  the  following,  which  is  an  accepted 
short  method  of  determining  the  average  of  a  number  of  measures  : 
"Arrange  the  numbers  in  tin1  order  <>!'  their  amount;  choose  any 
number  likely  to  be  uearesl  the  average ;  a<l<l  together,  regarding 
signs,  the  deviations    from    it   of  all    the  numbers  J   divide  this  result 


•I  MENTAL   AND  social   MEASUREMENTS 

by  the  Dumber  of  the  measures  the  average  of  which  you  are  ob- 
taining: add  the  quotient  to  the  chosen  number."  To  secure  full 
mastery  of  every  procedure  taught,  this  book  will  contain  many 
model  examples  and  sets  of  problems  to  be  worked. 

The  Application  of  the  Tlieory  of  Measurements. 
A  -ense  of  when  aud  how  to  use  statistical  methods  is  even  more 
important  than  knowledge  of  the  methods  themselves.  The  greatest 
benefit,  therefore,  will  come  to  those  who  in  connection  with  every 
principle  established  in  the  text,  call  to  mind  some  concrete  case  to 
which  the  principle  should  be  applied.  The  insight  into  the  actual 
use  of  the  theory  of  measurement  thus  obtained  may  be  increased  by 
a  critical  examination  of  the  samples  of  quantitative  studies  referred 
to  in  Chapter  XIII. 

The  Theory  of  Measurements  and  the  Special  Sciences. 
This  book,  as  the  title  announces,  deals  primarily  with  the  theory 
of  mental  measurements.  But  with  a  few  exceptions  the  principles 
and  technique  which  it  presents  are  applicable  to  all  the  sciences 
which  study  variable  phenomena.  So  far,  indeed,  physical  anthro- 
pology has  been  the  science  to  take  the  most  advantage  of  them,  and 
in  medicine  they  will  perhaps  find  their  greatest  usefulness.  The 
illustrations  occasionally,  and  the  problems  frequently,  come  from  the 
biological  sciences.  If  one  alters  the  language  and  replaces  the 
illustrations  from  the  realms  of  psychology  and  social  science  by 
similar  ones  from  economics,  vital  statistics,  medicine,  physiology, 
anthropometry  or  biology,  as  the  case  may  be,  he  will  find  the  prin- 
ciples to  hold,  with  an  occasional  obvious  modification  to  fit  the 
special  data.  The  descriptions  of  technical  procedure  similarly  may, 
after  a  few  obvious  alterations,  be  applied  to  variable  measurements 
in  general. 

The  Intrinsic  Interest  of  the  Theory  of  Measurements. 

The  author  may  be  permitted  to  express  his  hope  that  those  who 
use  the  book  will  regard  its  subject  matter  as  something  more  than 
a  means  to  the  end,  convenient  handling  of  measurements.  One 
can  use  ingenuity  in  manipulating  measurements  as  well  as  in  devis- 
ing experiments ;  can  use  logic  in  working  with  measures  as  well  as 
in  working  wTith  evidence  of  a  more  impressive  and  dramatic  sort. 


INTRODUCTION  5 

Skill  in  expression  is  nowhere  more  required  than  in  the  task  of 
making  quantitative  estimates,  comparisons  and  relationships,  brief, 
clear  and  emphatic.  Statistics  are,  or  at  least  may  be,  something 
beyond  tabulation  and  book-keeping.  In  studying  even  this  most 
elementary  introduction  one  who  is  willing  to  use  his  higher  intel- 
lectual powers  will  find  something  for  them  to  do. 

The  Special  Problems  of  Mental  Measurements. 

In  the  mental  sciences  as  in  the  physical  we  have  to  measure 
things,  differences,  changes  and  relationships  or  dependencies.  The 
psychologist  thus  measures  the  acuity  of  vision,  the  changes  in  it 
due  to  age,  and  the  relationship  between  acuity  of  vision  and  ability 
to  learn  to  spell.  The  economist  thus  measures  the  wealth  of  a 
community,  the  changes  due  to  certain  inventions  and  perhaps  the 
dependence  of  the  wealth  of  communities  upon  their  tariff  laws  or 
labor  laws  or  poor  laws.  Such  measurements,  which  involve  human 
capacities  and  acts,  are  subject  to  certain  special  difficulties,  due 
chiefly  to  the  absence  or  imperfection  of  units  in  which  to  measure, 
the  lack  of  constancy  in  the  facts  measured  and  the  extreme  com- 
plexity of  the  measurements  to  be  made. 

If,  for  instance,  one  attempts  to  measure  even  so  simple  and  me- 
chanical a  thing  as  the  spelling  ability  of  ten-year-old  boys,  one  is 
hampered  at  the  start  by  the  fact  that  there  exist  no  units  in  which 
to  measure.  One  may,  of  course,  arbitrarily  make  up  a  list  of  10 
or  50  or  100  words  and  measure  ability  by  the  number  spelled  cor- 
rectly. But  if  one  examines  such  a  list,  for  instance  the  one  used  by 
Dr.  J.  M.  Rice  in  his  measurements  of  the  spelling  ability  of  18,000 
children,  one  is  or  should  be  at  once  struck  by  the  inequality  of  the 
units.  Is  '  to  spell  certainly  correctly '  equal  to  '  to  spell  because 
correctly'?  In  point  of  fact,  I  find  that  of  a  group  of  about  120 
children,  30  missed  the  former  and  only  one  the  latter.  All  of  Dr. 
Rice's  results  which  are  based  on  the  equality  of  any  one  of  his  50 
words  with  any  other  of  the  50  are  necessarily  inaccurate,  as  is 
abundantly  shown  by  Table  I.  (page  8). 

Economists  have  not  yet  agreed  upon  a  system  of  units  of  meas- 
urement of  consuming  power.     Is  an  adult  man  to  be  scored  as  twice* 
or  two  and  a  half  or  three  times  as  great  a  consumer  as  a  ten-year- 
old  boy?     If  an  adult  man's  consuming  power  equals  1.00,  what  is 
the  value  of  that  of  an  adult  woman  ? 


6  MENTAL   AND  SOCIAL    MEASUREMENTS 

[fwe  measure  a  school  hoy's  memory  or  a  school  system's  daily 
attendance  or  a  working  man's  daily  productiveness  or  a  family's 
daily  expenditures,  we  find  in  any  case  not  a  single  result,  hut  a  set 
of  varying  results.  The  force  of  gravity,  the  ratio  of  the  weight  of 
( )  to  the  weight  of  11  id  water,  the  mass  of  the  H  atom,  the  length 
of  a  given  wire  ;  these  are,  we  say,  constants;  and  though  in  a  series 
of  measures  we  get  varying  results,  the  variations  are  very  slight  and 
can  be  attributed  to  the  process  of  measuring.  But  with  human  af- 
fairs not  only  do  our  measurements  give  varying  results  ;  the  thing 
itself  is  not  the  same  from  time  to  time,  and  the  individual  things  of 
a  common  group  are  not  identical  with  each  other.  If  we  say  that 
the  mass  of  the  O  atom  is  16  times  the  mass  of  the  H  atom,  we  mean 
that  it  always  is  that  or  very,  very  near  it.  But  if  we  say  that  the 
size  of  the  American  family  is  2  children,  we  do  not  mean  that  it  is 
that  alone  ;  we  mean  that  it  is  sometimes  0,  sometimes  1,  etc. 

Even  a  very  elaborate  chemical  analysis  would  need  only  a  score 
or  so  of  different  substances  in  terms  of  which  to  describe  and  meas- 
ure its  object,  but  even  a  very  simple  mental  trait,  say  arithmetical 
ability  or  superstition  or  respect  for  law,  is,  compared  with  physical 
things,  exceedingly  complex.  The  attraction  of  children  to  certain 
studies  can  be  measured,  but  not  with  the  ease  with  which  we  can 
measure  the  attraction  of  iron  to  the  magnet.  The  rise  and  fall  of 
stocks  is  due  to  law,  but  not  to  any  so  simple  a  law  as  explains  the 
rise  and  fall  of  mercury  in  a  thermometer. 

The  problem  for  a  quantitative  study  of  the  mental  sciences  is 
thus  to  devise  means  of  measuring  things,  differences,  changes  and 
relationships  for  which  standard  units  of  amount  are  often  not  at 
hand,  which  are  variable,  and  so  unexpressible  in  any  case  by  a  sin- 
gle figure,  and  which  are  so  complex  that  to  represent  any  one  of 
them  a  long  statement  in  terms  of  different  sorts  of  quantities  is  com- 
monly needed.  This  last  difficulty  of  mental  measurements  is  not, 
however,  one  which  demands  any  form  of  statistical  procedure  essen- 
tially different  from  that  used  in  science  in  general. 


CHAPTER   II. 

UNITS    OP    MEASUREMENT. 

Let  us  examine  first  a  number  of  units  that  have  actually  been 
used.  It  is  the  custom  to  measure  intellectual  ability  and  achieve- 
ment, as  manifested  in  school  studies,  by  marks  on  an  arbitrary 
scale;  for  instance,  from  0  to  100  or  from  0  to  10.  Suppose  now 
that  one  boy  in  Latin  is  scored  60  and  another  90.  Does  this  mean, 
as  it  would  in  ordinary  arithmetic,  that  the  second  boy  has  one  and 
one  half  times  as  much  ability  or  has  done  one  and  one  half  times 
as  well  ?  It  may  by  chance  in  some  cases,  but  the  fact  that  the 
best  one  and  the  worst  one  of  thirty  boys  may  be  so  marked  by 
one  teacher,  and  during  the  next  half  year  in  the  same  study  be 
marked  70  and  90  by  the  next  teacher,  proves  that  it  need  not.  The 
same  difference  in  ability  may,  in  fact,  be  denoted  by  the  step  from 
60  to  90  by  one  teacher,  by  the  step  from  40  to  95  by  another,  by 
the  step  of  from  75  to  92  by  another  and  even  by  still  another  by 
the  step  from  90  to  96.  Obviously  school  marks  are  quite  arbitrary 
and  their  use  at  their  face  value  as  measures  is  entirely  unjustifiable. 
A  90  boy  may  be  four  times  or  three  times  or  six  fifths  as  able  as 
an  80  boy. 

It  is  the  custom  to  measure  the  value  of  commodities  and  labor 
by  their  money  price,  but  since  a  dollar  in  one  year  is  evidently  not 
necessarily  equal  to  a  dollar  twenty  years  before,  systems  of  index 
values  *  have  been  established  to  give  a  better  unit.  Even  these 
index  values  as  arranged  by  different  statisticians  differ  somewhat. 

For  a  unit  of  power  of  consumption  Engel  takes  a  child  during 
its  first  year.  He  then  calls  a  year-old's  power  of  consumption  1.1  ; 
a  two-year-old's,  1.2  ;  and  so  on  up  to  3.0  for  a  woman  20  years  or 
over  and  3.5  for  a  man  25  years  or  over.  In  the  United  States 
investigation  of  1890-91  the  unit  was  taken  as  100  for  an  adult 
man,  90  for  an  adult  woman,  75  for  a  child  7  to  10  years  old,  40 
for  a  child  3  to  6  and  15  for  a  child  1  to  3.  The  arbitrary  nature 
of  the  scale  of  measurement  is  apparent. 

*The  reader  unlearned  in  economic  science  may  neglect  this  illustration. 

7 


8  MENTAL    A\D  SOCIAL    Ml-ASTREMEXTS 

The  extreme  inequalities  of  the  spelling  words,  treated  by  Dr. 
lviiv  as  of  equal  difficulty,  are  shown  in  Table  I. 


TABLE  I. 

The  Rej  uivk  Frequency  of  Mistakes  within  the  Same  Group  of 

Children  for  Each  of  49  "Words  taken  by  Dr.  Rice  to  be  of 

Equal  Amount  as  Measures  of  Spelling  Ability. 


By  5"  Grade 

By  5«  Grade 

By  5"  Grade 

By  5«  Grade 

Girls. 

Boys. 

Girls. 

Boys. 

I  disappoint 

24 

13 

Frightened 

3 

6 

Necessary 

23 

19 

Baking 

3 

6 

Changeable 

20 

22 

Peace* 

3 

6 

Almanac 

19 

14 

Laughter 

3 

6 

Certainly 

15 

15 

Waiting 

2 

8 

Lose 

15 

12 

Chain 

2 

7 

Slipped 

13 

9 

Thought 

2 

6 

Deceive 

13 

7 

Weather 

2 

4 

Whistling 

11 

11 

Light 

2 

4 

Purpose 

9 

10 

Surface 

2 

4 

Speech 

8 

15 

Strange 

2 

4 

Receive 

7 

12 

Enough 

2 

2 

Loose 

7 

7 

Running 

2 

2 

Listened 

6 

9 

Distance 

i 

6 

Choose 

6 

6 

Getting 

1 

3 

Queer 

6 

5 

Better 

1 

2 

Hopping 

6 

5 

Feather 

1 

0 

Believe 

5 

8 

Rough 

0 

5  . 

Writing 

5 

7 

Covered 

0 

5 

Smooth 

5 

5 

Always 

0 

4 

Language 

5 

3 

Mixture 

0 

4 

Neighbor 

4 

7 

Driving 

0 

3 

Learn 

4 

2 

Because 

0 

1 

Changing 

3 

11 

Picture 

0 

0 

Careful 

3 

8          1 

In  the  three  cases  so  far  the  arbitrary  opinions  or  guesses  of 
individuals  that  such  and  such  are  equal  have  been  uncritically  ac- 
cepted. It  is  as  if  we  should  measure  length  in  accord  with  some 
one's  guess  that  the  distance  from  San  Francisco  to  Chicago  equaled 
three  times  that  from  Chicago  to  New  York  and  eight  times  that 
from  Xew  York  to  Boston. 

The  risk  of  accepting  subjective  opinion  even  in  the  cases  where 
it  is  least  liable  to  error  may  be  illustrated  by  the  variation  in  judg- 
ment, even  among  competent  authorities  (graduate  students  of  ex- 
perience in  teaching),  as  to  the  relative  difficulty  of  different  parts  of 
the  following  simple  tests  : 

*  Piece  was  scored  correct. 


UNITS  OF  MEASUREMENT 


144      27       2      27 
How  much  is  — ^-  X  —  x  ^  x  j^  ? 

How  much  is  5f  +  1£  —  7£  +  6|  ? 

3.  If  a  girl  had  two  dollars,  three  five-cent  pieces,  two  dimes 
and  three  quarter  dollars,  how  much  money  would  she  have  in  all  ? 

250  1 

4.  How  much  is  374  +  87J  4-  — =-  +  6  +  ^4-  6  ? 

Twelve  individuals  assigned  to  examples  2,  3  and  4  the  amount 
of  credit  due  for  successful  solution  of  each  on  the  basis  that  the  suc- 
cessful solution  of  example  1  received  a  credit  of  10.  They  esti- 
mated, that  is,  the  abilities  involved  in  doing  2,  3,  and  4  in  terms 
of  the  ability  involved  in  doing  1.  Their  estimates  varied  from  8 
to  20  for  2,  from  5  to  20  for  3,  and  from  14  to  25  for  4.  Their 
ratings  in  detail  were  (Table  II.): 

TABLE  II. 

Example  2.  Example  3.  Example  4. 

Rating.    Number  giving  it.    Rating.  Number  giving  it.      Rating.    Number  giving  it. 

8  1  5  5  13  1 

10  1  6  1  14  1 

12  1  8  1  15  4 

15  6  10  1  18  2 

18  1  12  1  20  3 

20  2  15  2  25  1 

20  1 

These  variations  are  due  to  two  factors ;  first,  the  variations  in 
the  opinions  of  the  difficulty  of  the  standard  (example  1)  and,  sec- 
ond, the  variations  in  the  opinions  of  the  difficulty  of  2,  3  and  4. 
We  may  eliminate  the  first  factor  and  measure  the  variation  which 
would  appear  if  the  different  individuals  compared  their  opinions  of 
2,  3  and  4  with  some  objective  standard  by  dividing  their  ratings  for 
each  single  example  by  the  average  of  their  ratings  for  all  three. 
When  this  is  done  their  estimates  still  range  from  6.7  to  13.7  for  2, 
from  3.0  to  10.9  for  3,  and  from  10.0  to  15.5  for  4. 

So,  also,  if  we  take  four  individuals  whose  ratings  were  such  as 
to  show  that  they  were  practically  identical  in  their  estimates  of  the 
difficulty  of  1,  we  find  that  even  among  just  these  four  the  ranges  are 
10  to  20,  5  to  15  and  15  to  25  for  2,  3  and  4  respectively. 


10  MENTAL   AND  SOCIAL    MEASUREMENTS 

The  detailed  corrected  ratings  were  : 

Example  2.  6.7,  8.2,  8.3,  9.0,  10.6,  10.9,  11.3,  12.0,  12.9,  12.9,  13.3,  13.7. 
•  3.  3.0,  3.S,  4.3,  4.3,  4.4,  4.5,  5.9,  6.2,  6.7,  9.0,  10.0,  10.9. 
■       4.     10.0,  10.9,  10.9,  11.2,  12.0,  12.4,  12.9,  12.9,  13.2,  13.3,  15.0,  15.5. 

The  percentages  of  highest  to  lowest  ratings  of  the  three  examples 
are  thus  245,  303  and  155.  If  we  choose  the  closest  limits  which 
will  include  8  out  of  the  12  ratings,  the  percentages  that  the  upper  is 
oi'  the  lower  limits  are  :  for  example  2,  129  ;  for  3,  176  ;  and  for  4, 
122. 

B. 

Write  as  quickly  as  you  can  besides  each  word  in  the  column  a 
word  that  means  the  opposite  thing  from  it. 

1.  Vertical. 

2.  Ignorant. 

3.  Kude. 

4.  Simple. 

5.  Deceitful. 

6.  Stingy. 

7.  Permanent. 

8.  Over. 

9.  To  degrade. 

10.  Weary. 

11.  To  spend. 

12.  To  reveal. 

13.  Genuine. 

14.  Level. 

15.  Broken. 

16.  Wild. 

17.  Part. 

18.  Past. 

19.  Permit. 

20.  Precise. 

Eight  individuals  assigned  to  words  2  to  20  the  amount  of  credit 
due  for  correctly  writing  the  opposite  of  each  of  them,  on  the  basis 
that  the  credit  for  writing  the  opposite  to  word  1  should  be  arbi- 
trarily called  10.  Their  estimates  varied  very  widely,  as  may  be 
seen  from  the  table  (III.)  below  : 

Of  course,  as  in  case  A,  some  of  this  variation  is  due  to  the 
varying  opinions,  of  the  difficulty  of  thinking  of  the  opposite  of 
the  first  word  vertical.  Any  one  word  would  be  an  insufficient  test. 
The  influence  of  subjective  opinion  is,  therefore,  more  fairly  meas- 
ured by  using  only  those  individuals  whose  ideas  of  the  difficulty  of 


UNITS   OF  MEASUREMENT 


11 


TABLE  III. 


Word. 


Ignorant 

Kude 

Simple 

Deceitful 

Stingy 

Permanent 

Over 

To  degrade 

Weary 

To  spend 

To  reveal 

Genuine 

Level 

Broken 

Wild 

Part 

Past 

Permit 

Precise 


Detailed  Credits  Given. 


5, 

6, 

8, 

8, 

10, 

11, 

15, 

6, 

7, 

7, 

9, 

10, 

13, 

14, 

2, 

4, 

8, 

9, 

12, 

14 

15, 

2, 

6, 

8, 

10, 

10, 

15, 

18, 

5, 

8, 

9, 

9, 

10 

12 

12, 

9, 

9, 

10, 

10 

10 

11 

14, 

1, 

2, 

6, 

6, 

6 

8 

8, 

2, 

5, 

7, 

10 

10 

14 

15, 

6, 

7, 

9, 

10 

13 

15 

18, 

2, 

5, 

6, 

8 

9 

10 

11, 

2, 

8, 

8, 

10 

10 

11 

!2, 

7, 

9, 

9, 

11 

12 

12 

15, 

8, 

9, 

9, 

10 

10 

15 

15, 

6, 

8, 

8, 

8, 

8, 

10 

15, 

2, 

6, 

6, 

7, 

8 

8 

9, 

2, 

5, 

7, 

7, 

8, 

9 

10, 

2, 

6, 

7, 

8, 

8 

9 

10, 

8, 

9, 

9, 

11, 

11, 

12, 

15, 

10, 

11, 

11, 

11 

18, 

20 

20, 

15. 
15. 
18. 
25. 
20. 
15. 
8. 
20. 
20. 
12. 
15. 
16. 
18. 
18. 
15. 
20. 
30. 
15. 
20. 


the  standard  were  alike,  or  by  allowing  for  the  differences.  Divid- 
ing the  ratings  of  each  individual  by  the  average  of  all  his  ratings, 
Table  III.  becomes  Table  IV.  Table  IV.  contains  also  measures 
of  the  range  in  terms  of  the  percentage  that  the  upper  is  of  the 
lower  limit  in  the  case  of  each  word. 


Word.    Range  of  Credits 
Given. 

Ignorant     6.6-11.6     6.6 

7.0 

TABLE  IV. 

Detailed  Credits  Given. 
7.6    7.9      9.9     10.7 

11.1 

11.6 

Per  cent 
Highest 

is  of 
Lowest. 

176 

Per  cent. 
Upper  is 
of  Lower 
of  Limits 
Including 
5  Ratings 

147 

Rude 

6.9-12.5 

6.9 

7.9 

8.9  10.0 

10.1 

11.4 

12.3 

12.5 

181 

125 

Simple 

3.6-13.6 

3.6 

5.3 

8.9    9.6 

10.9 

11.1 

11.9 

13.6 

378 

134 

Deceitful 

3.6-15.9 

3.6 

7.9 

7.9  11.1 

11.4 

12.3 

14.0 

15.9 

442 

143 

Stingy 

6.6-16.1 

6.6 

8.9 

9.1     9.3 

9.9 

11.1 

12.7 

16.1 

244 

125 

Permanent  6. 4-16. 1 

6.4 

8.5 

11.1  11.4 

11.8 

12.3 

13.9 

16.1 

252 

125 

Over 

1.8-  8.9 

1.8 

2.6 

3.8    5.9 

6.1 

6.2 

7.4 

8.9 

495 

151 

To  degrad 

e  3.6-15.5 

3.6 

6.4 

6.6    8.6 

11.1 

11.4 

13.9 

15.5 

431 

178 

Weary 

9.2-14.0 

9.2 

10.0 

10.7  11.4 

12.3 

12.7 

12.9 

14.0 

152 

123 

To  spend 

3.6-11.1 

3.6 

5.9 

6.1     6.6 

7.6 

8.5 

11.1 

11.1 

308 

144 

To  reveal 

3.6-14.9 

3.6 

7.6 

7.6    8.5 

9.9 

10.5 

11.1 

14.9 

414 

138 

Genuine 

7.6-16.1 

7.6 

8.6 

9.3  10.0 

11.4 

14.5 

15.8 

16.1 

212 

150 

Level 

9.6-17.9 

9.6 

9.9 

9.9  10.0 

11.6 

11.8 

13.6 

17.9 

186 

119 

Broken 

7.4-14.3 

7.4 

7.6 

7.9    8.9 

10.5 

11.4 

11.5 

14.3 

193 

142 

Wild 

3.6-  9.6 

3.6 

5.9 

6.1     7.0 

7.9 

8.6 

8.9 

9.6 

267 

137 

Part 

3.6-12.7 

3.6 

G.2 

6.9     7.0 

7.6 

8.9 

9.2 

12.7 

353 

133 

Past 

3.6-19.1 

3.6 

6.0 

6.9     7.4 

8.5 

8.8 

11.8 

19.1 

531 

147 

Permit 

7.0-19.6 

7.0 

9.6 

9.9  10.0 

10.9 

11.4 

15.8 

19.6 

280 

119 

Precise 

11.1-26.3 

11.1 

11.5 

13.6  13.9 

15.2 

15.5 

25.0 

26.3 

237 

135 

12  MENTAL  AND  social  MEASUREMENTS 

On  tlu'  average  the  highest  rating  is  three  times  the  lowest,  and 
the  upper  of  the  limits,  including  five  ratings  out  of  the  eight,  is  one 
and  tlnve  eighths  times  the  lower. 

C. 
Write  beside  each  of  these  words  a  word  which  means  some  kind 
of  the  thing-  that  the  printed  words  means. 

1.  Musician 

2.  Official 

3.  Criminal 

4.  Fish 

5.  Game 

6.  Study 

7.  Machine 

8.  Building 

9.  Furniture 

10.  Fruit 

11.  Clothes 

12.  Vegetable 

13.  Book 

14.  Boat 

15.  Tree 

16.  Dish 

17.  Plant 

18.  Timepiece 

19.  Disease 

20.  Pain 

21.  Part  of  speech 

22.  Superior  officer 

Seven  individuals  assigned  to  words  2  to  22,  the  amount  of 
credit  due,  in  their  opinion,  for  correctly  writing  a  corresponding 
species  word  for  each  of  them,  on  the  basis  that  the  credit  for  writing  a 
word  naming  a  kind  of  musician  should  be  called  10.  In  this  case 
I  will  give  only  the  estimates  so  corrected  as  to  eliminate  differences 
of  opinion  with  respect  to  the  difficulty  of  word  1,  and  will  include, 
as  in  Table  IV.,  the  percentages  of  highest  to  lowest  ratings  and 
the  percentages  that  the  upper  is  to  the  lower  of  the  limits  that  in- 
clude five  out  of  the  seven  ratings. 

On  the  average  the  highest  rating  is  a  trifle  over  two  and  three 
quarters  times  the  lowest,  and  the  upper  of  the  limits  including  five 
ratings  is  almost  one  and  one  half  times  the  lower. 

In  college  registration  statistics  the  unit  taken  is  commonly  one 
student.     The  college  with  a  score  of  400  is  supposed  to  be  twice  as 


UNITS  OF  MEASUREMENT 


13 


Word. 
Official 

Range. 
8.1-15.9 

8.1 

TABLE  V. 

Detailed  Credits  Given. 

10.0  12.1  12.1     14.7 

15.5 

Per  cent. 
Highest  is 
of  Lowest. 

15.9     196 

Per  cent. 
Upper  is 
of  Lower 
of  Limits 
Including 
5  Eatings. 

131 

Criminal 

6.1-21.2 

6.1 

10.0 

12.1 

12.3 

15.9 

18.1 

21.2 

348 

175 

Fish 

4.8-14.5 

4.8 

6.1 

10.0 

10.3 

10.6 

11.1 

14.5 

302 

145 

Game 

4.8-13.3 

4.8 

8.0 

8.1 

9.0 

9.1 

10.0 

13.3 

277 

125 

Study 

4.8-10.9 

4.8 

6.0 

6.4 

8.1 

8.6 

10.0 

10.9 

227 

167 

Machine 

9.0-15.9 

9.0 

10.0  10.2 

12.1 

13.5 

14.5 

15.9 

177 

145 

Building 

6.0-11.1 

6.0 

7.9 

9.0 

9.1 

9.7 

10.0 

11.1 

185 

123 

Furniture 

6.0-23.8 

6.0 

7.2 

7.7 

8.1 

9.8 

10.0 

23.8 

397 

139 

Fruit 

4.5-15.9 

4.5 

6.0 

7.4 

7.7 

8.1 

10.0 

15.9 

353 

167 

Clothes 

4.5-12.2 

4.5 

6.0 

6.1 

6.4 

7.9 

10.0 

12.2 

271 

167 

Vegetable 

6.0-15.9 

6.0 

7.2 

7.4 

7.7 

9.1 

10.0 

15.9 

265 

139 

Book 

4.8-11.1 

4.8 

6.0 

6.4 

9.1 

10.0 

10.2 

11.1 

231 

170 

Boat 

3.2-18.0 

3.2 

4.8 

10.0 

10.3 

12.3 

13.2 

18.0 

563 

180 

Tree 

3.2-10.0 

3.2 

6.1 

7.7 

9.1 

9.7 

9.8 

10.0 

313 

130 

Dish 

3.2-10.0 

3.2 

7.2 

7.4 

7.6 

9.0 

9.1 

10.0 

313 

126 

Plant 

4.8-14.5 

4.8 

6.0 

8.6 

9.0 

10.0 

10.2 

14.5 

302 

169 

Timepiece 

6.0-10.0 

6.0 

6.4 

7.2 

7.4 

7.9 

9.1 

10.0 

167 

132 

Disease 

7.7-15.9 

7.7 

9.1 

9.8 

10.0 

12.1 

13.2 

15.9 

206 

145 

Pain 

7.9-22.7 

7.9 

9.0 

10.0 

10.9 

12.3 

15.2 

22.7 

287 

127 

Part  of  speech 

6.0-11.6 

6.0 

6.4 

7.2 

8.6 

10.0 

10.2 

11.6 

193 

159 

Superior  officer  10.0-25.9 

10.0 

11.1 

16.9 

18.1 

19.1 

20.3 

25.9 

259 

153 

large  as  the  college  with  200.  But  some  students  do  four  years' 
work  in  three,  some  are  present  only  a  part  of  the  year  or  take  only 
a  fraction  of  the  full  course  during  their  time  of  enrollment.  A 
university  with  1,000  units  made  up  in  part  of  teachers  taking  a 
course  or  two  a  year,  of  casual  students  that  drop  out  to  take  posi- 
tions and  of  other  irregulars,  might  really  have  a  smaller  attendance 
in  the  true  sense,  a  smaller  influence  on  students,  than  one  with  only 
800  units.  One  person  equals  one  person  as  a  name  or  physical 
unit,  but  one  person  studying  all  his  time  with  regular  and  continued 
attendance  does  not  equal  one  person  taking  university  work  as  a 
secondary  pursuit. 

In  measuring  the  fertility,  or  rather  the  reproductivity  of  human 
beings,  it  seems  at  first  thought  to  be  justifiable  to  use  the  number 
of  children  in  the  family  as  a  measure.  But  is  not  the  number  of 
children  who  live  a  better  measure?  And  may  not  the  number  of 
children  who  live  through  the  reproductive  period  (say  50  years)  be 
a  still  better  measure,  and  is  not  perhaps  the  number  of  children, 
each  weighted  in  some  way  by  the  length  of  his  life,  another  measure 


ll  MENTAL   AND  SOCIAL   MEASUREMENTS 

to  be  considered?  Surely  a  child  who  dies  in  five  minutes  is  not 
equal  as  a  measure  of  reproductivity  to  a  child  who  lives  sixty 
years,      [a  a  child  who  lives  only  thirty  years? 

In  the  case  of  the  '  college  student '  and  the  'child  born'  we  are 
misled  by  what  Professor  Aikins  has  called  the  'jingle'  fallacy. 
The  words  are  identical  ami  we  tend  to  accept  all  the  different 
things  t"  which  they  may  refer  as  of  identical  amount.  A  similar 
unthinking  acceptance  of  verbal  equality  as  a  proof  of  real  equality 
makes  one  measure  labor  on  the  hypothesis  that  any  one  hour  is 
equal  to  any  other  hour  of  it,  forgetting  that  the  step  from  7  to  8 
hours  per  diem  may  be  quite  different  from  the  step  from  8  to  9  and 
is  obviously  far  different  from  the  step  from  20  to  21  hours.  The 
fallacy  may  be  emphasized  by  one  final  illustration.  Dr.  Swift,  in 
studving  the  effect  of  practice,  measured  motor  skill  by  the  number 
of  time  two  balls  could  be  kept  tossed  in  the  air  with  one  hand.  He 
took  as  a  unit  of  measurement  one  successful  pair  of  tosses  and  re- 
garded any  one  such  pair  as  equal  to  any  other.  For  him,  that  is, 
the  step  from  0,  or  inability  to  catch  and  toss  again  at  all,  to  5,  or 
the  ability  to  catch  and  toss  5  times  with  each  ball,  is  equal  to  the 
Btep  from  200,  or  ability  to  keep  the  balls  in  the  air  200  times  with- 
out failure,  to  205,  or  the  ability  to  do  so  205  times.  But,  of  course, 
if  one  can  do  the  performance  200  times  he  can,  so  far  as  motor 
skill  goes,  do  it  205  times  almost  as  easily,  the  step  being  nearly 
zero.  On  the  other  hand,  the  step  from  0  to  5  is  a  very  consider- 
able gap,  one  which  some  individuals  can  never  pass.  The  result 
of  Dr.  Swift's  system  of  units  is  that  he  gets  the  appearance  of  very 
slow  improvement  in  early  hours  of  practice  and  very  rapid  improve- 
ment in  late  hours,  a  state  of  affairs  which  contradicts  what  is  found 
by  other  investigators.  Of  course,  '  tossing  two  balls  once '  sounds 
identical  with  '  tossing  two  balls  once,'  but  it  is  not. 

In  arranging  a  scale  of  measurement  one  must  so  far  as  possible, 
(1)  keep  free  of  individual  opinion,  must,  i.  e.,  be  supported  by  the 
agreement  of  all  qualified  observers.  This  is  most  satisfactorily  ac- 
complished by  so  arranging  observations  or  experiments  that  the 
trait  is  measured  in  terms  of  some  objective  units,  such  as  seconds, 
millimeters,  dollars.  Thus,  ability  to  memorize  can  be  measured 
by  time  taken  more  justly  than  by  amount  done,  for  a  second  is  a 
second,  while  one  line  of  poetry  may  be  easier  than  another  line. 


UNITS  OF  MEASUREMENT  15 

The  accuracy  of  movement  as  tested  by  attempts  to  hit  a  dot  can  be 
measured  more  justly  by  actual  measurement  than  by  mere  inspec- 
tion ;  men  can  be  ranked  as  to  wealth  better  by  valuations  of  their 
property  than  by  the  opinions  of  their  neighbors.  One  must  also 
(2)  call  equal  only  those  things  which  can  be  interchanged  without 
making  any  difference  to  the  issue  involved.  Twelve  inches  can  be 
thus  interchanged  with  one  foot  without  making  any  difference  if 
the  issue  is  physical  measurement,  but  not  if  it  is  the  study  of  lan- 
guage. Ten  dimes  can  be  thus  interchanged  with  one  dollar  if  the 
issue  is  the  accounts  of  a  store,  but  not  if  it  is  the  area  of  surfaces. 

Even  where  there  are  available  units  of  amount  which  are  com- 
mensurable they  are  rarely  on  a  scale  with  a  known  zero  point. 
Measurements  of  the  time  taken  to  hear  a  sound  and  react  by  lift- 
ing a  finger  are  commensurable  in  the  sense  that  140  is  as  much 
faster  than  150  as  150  is  than  160,  but  an  absolute  zero  point  for 
slowness  is  not  known.  It  is  impossible,  then,  to  argue  about 
quickness  of  reaction  as  we  can  about  mass  or  temperature. 

The  ability  to  spell  correctly  disappoint  and  almanac  may  be 
found  to  be  equal  to  the  ability  to  spell  correctly  necessary  and 
changeable,  but  how  much  of  an  advance  it  is  beyond  the  absolute 
zero  of  spelling  ability  can  not  be  stated,  since  that  absolute  zero  is 
unknown.  It  may  be  taken  to  be  the  ability  to  spell  no  word  at  all. 
But  at  once  the  objection  is  raised  that  of  the  many  who  could  spell 
no  word  at  all  some  could  do  so  with  a  little  training,  while  some 
would  need  more,  and  a  few  among  the  idiots  could  never  with  all 
possible  training  be  gotten  to  spell  any.  In  physical  science  we  can 
find  or  infer  the  place  where  a  given  quantity  begins,  —  the  first  in- 
crement to  the  absolute  zero  of  temperature,  the  least  quantity  of  mass 
or  velocity  or  light,  the  least  degree  of  resistance,  etc.;  but  this  is 
rarely  our  good  fortune  when  dealing  with  mental  facts. 

The  zero  points  from  which  to  reckon  amounts  of  goodness,  in- 
tellect, delicacy  of  discrimination,  memory,  courage,  efficiency,  quick- 
ness, economic  productivity,  inventiveness,  etc.,  are  largely  lacking. 
Two  pounds  is  twice  one  pound  not  only  in  the  sense  that  it  takes 
two  of  the  latter  to  replace  one  of  the  former,  but  also  in  the  sense 
that  the  former  represents  a  point  on  the  scale  of  mass  twice  as  far 
from  the  zero  point  as  does  the  former.     Marking  20  A's  instead  of 


16  MENTAL    AND  SOCIAL  MEASUREMENTS 

10  on  a  sheet  of  mixed  capital  letters,  or  earning  $10.00  instead  of 
$5.00,  or  remembering  six  words  instead  of  three,  or  inventing  four 
machines  instead  of  two,  can  by  proper  choice  of  units  be  made  to 
parallel  the  two-pounds-one-pound  comparison  in  its  first  sense,  but 
not  in  its  second.  For  there  is  a  less  perceptive  ability  than  that  of 
just  barely  not  perceiving  any  A ;  productiveness  runs  into  minus 
quantities  in  the  case  of  workmen  who  spoil  raw  materials  with  no 
advantageous  result ;  there  are  lower  grades  of  memory  and  of  in- 
ventiveness than  those  of  just  not  remembering  one  word  or  of  just 
nut  inventing  one  thing. 

Even  when  absolute  zero  points  are  not  discoverable  it  is  well 
worth  while  to  consider  from  what  point  the  scale  we  do  use  starts  ; 
and  even  when  the  point  has  to  be  chosen  arbitrarily  it  is  well  worth 
while  to  consider  the  meaning  and  utility  of  different  possible  ones. 
It  is  the  duty  of  the  student  of  mental  and  social  quantities  to  study 
the  whole  scale  in  which  the  units  he  uses  lie,  as  well  as  to  turn 
those  he  does  use  into  conirnensurable  quantities. 

The  influence  of  the  zero  point  of  a  scale  upon  measurements 
made  by  that  scale  will  alter  the  interpretation  of,  but  not  the 
method  of  making,  measurements  of  things  and  conditions ;  but  when 
things  or  conditions  are  compared,  that  is,  when  measurements  are 
made  of  difference,  change  and  relationships,  it  becomes  of  the  ut- 
most importance.  For  one  of  the  common  fallacies  in  the  mental 
sciences  is  to  compare  directly  the  amounts  of  measurements  made 
from  different  zero  points.  Another  is  to  use  arbitrarily  some  point 
along  the  scale  as  if  it  were  an  absolute  zero  point.  Silly  as  it  may 
appear,  we  often  with  mental  measurements  do  such  arithmetic  as 
the  following  : 

"  John,  who  weighed  4  lbs.  more  than  100  lbs.,  has  added  2  lbs. 
to  his  weight;  James,  who  weighed  100  lbs.  more  than  10  lbs.,  has 
added  to  his  weight  50  lbs.  Both  gained  50  per  cent,  and  so  their 
relative  gains  were  equal." 

"John  weighs  10  lbs.  more  than  60  lbs.  James  weighs  2  lbs. 
more  than  60  lbs.     John  is  five  times  as  heavy  as  James." 

Quantities  to  be  measured  may  be  in  a  discrete  or  in  a  continuous 
series.  A  discrete  series  is  one  with  gaps.  Thus  if  we  measure  the 
number  of  children  in  a  class  we  can  get  only  integral  numbers. 


UNITS  OF  MEASUREMENT  17 

Sixth  tenths  of  a  man,  ninety-two  hundreds  of  a  man,  do  not  exist. 
There  is  a  gap,  between  one  man  and  two,  two  men  and  three,  etc. 
A  continuous  series,  such  as  time  or  velocity  or  intellect  or  wealth,  is 
in  theory  capable  of  any  degree  of  subdivision.  Almost  all  mental 
traits  and  social  facts  due  to  human  action  are  quantities  in  continu- 
ous series. 

Any  given  measure  of  a  continuous  series  means  not  a  single 
point  on  the  scale  of  measurement,  but  the  distance  along  that  scale 
between  two  limits.  Thus  if  we  measure  the  time  taken  to  perceive 
and  react  to  a  signal  in  thousandths  of  a  second  and  get  .143  sec.  as 
the  measure,  the  .143  means  commonly  that  that  was  the  nearest 
point,  that  the  time  was  nearer  to  .143  than  to  .142  or  to  .144  ;  and 
this  means,  of  course,  that  the  time  was  between  .1425  and  .1435. 
The  truer  statement  would  be,  '  A's  reaction  time  is  between  .1425 
and  .1435.'  If  we  measure  a  man's  wealth  in  dollars  as  73,448,  we 
do  not  mean  that  he  has  exactly  that,  but  that  that  is  the  nearest 
dollar  mark.  At  times  a  measure  does  not  mean  that  the  individual 
to  whom  it  is  given  is  nearer  to  that  measure  than  to  any  other  on 
the  scale  used,  but  that  he  is  above  it  and  not  up  to  the  next  meas- 
ure. For  instance,  if  a  boy  in  10  minutes  gets  the  answers  to  5 
problems  in  arithmetic,  we  would  commonly  score  him  5,  but  our  5 
would  mean,  'at  least  5  and  not  6.'  The  boy  might,  for  instance, 
have  almost  completed  the  sixth  in  his  mind,  and  really  be,  if  we  had 
a  finer  scale,  5.9.  In  mental  measurements,  then,  any  figure,  say 
21,  may  mean  between  20.5  and  21.5  or  between  21  and  22.  It 
might  also  mean  between  20  and  21,  if  we  measured  people  by  the 
point  which  they  just  did  not  reach,  but  this  is  almost  never  a  useful 
method.  The  second  method  of  measuring  by  the  last  point  on  the 
scale  passed  is  in  many  mental  traits  the  natural  one  and  often  saves 
labor  in  all  sorts  of  measurements.* 

In  later  operations  with  figures  denoting  measurements  the 
method  of  obtaining  them  and  their  consequent  meaning  must  be 
kept  in  mind.  If  a  set  of  measures  mean  in  each  case  '  from  this  fig- 
ure to  the  next  on  the  scale,'  then  the  average  calculated  from  them 
will,  to  represent  an  absolute  point  on  the  scale,  need  to  be  increased 

*  It  is  easier  to  put  a  measure  between  two  points  on  the  scale  than  to  tell  to 
which  point  it  is  nearest.  Moreover,  in  dropping  insignificant  figures  it  is  easier  to 
drop  absolutely  than  to  add  1  place  when  the  figure  dropped  is  over  .5  the  unit  of 
the  next  place. 


18  MENTAL   A\I>  SOCIAL   M i:\sri;i;Mh\\Ts 

1>\-  .5  the  unit  o\'  the  scale.  A  little  experimentation  and  thought 
will  create  in  one  the  useful  habits  of  thinking  of  any  figure  for  a 
measure  on  a  continuous  scale  as  representing  the  quantities  between 
tw.>  limit-  ;  of  realizing  that  for  our  ordinary  arithmetic  it  represents 
tin-  space  from  a  point  half-way  between  it  and  the  figure  below  to 
a  point  half-way  between  it  and  the  figure  above  ;  and  of  understand- 
ing that  if  our  method  of  measurement  makes  it  represent  some  other 
space,  we  must  make  proper  allowance  in  calculation. 

In  many  cases  the  measure  of  zero,  which  should  mean  a  definite 
distance  on  the  scale,  either  from  a  point  below  0  to  a  point  above 
it,  or  from  0  to  the  next  point  on  the  scale,  means  only  an  indefinite 
distance ;  namely,  from  a  point  above  0  to  an  unknown  lower  ex- 
treme. Thus  if  in  measuring  arithmetical  ability  by  a  test  of  20 
examples,  we  should  find  out  of  fifty  boys  a  dozen  who  did  none  at 
all  and  should  mark  them  zero,  we  could  not  assume  that  they  were 
as  a  group  the  same  distance  below  the  1  to  2  group  as  the  1  to  2 
group  were  below  the  2  to  3  group.  All  that  wrould  be  known 
about  the  dozen  boys  would  be  that  they  belonged  somewhere  below 
1.  One  of  them  might  be  really  as  far  below  a  boy  marked  1  as 
the  latter  was  below  a  boy  marked  20.  In  such  cases  we  call  the 
zero  marks  undistributed  or  indefinite.  The  same  holds  good,  of 
course,  for  the  upper  as  well  as  the  lower  extreme.  If,  in  the  illus- 
tration in  question,  a  dozen  boys  had  done  all  the  examples  perfectly 
and  been  marked  20,  that  score  would  mean,  not  that  the  boys  were 
between  20  and  21,  but  that  they  were  somewhere  above  20.  One 
should  always  guard  against  undistributed  measures  at  the  extreme 
of  a  scale. 

Many  mental  phenomena  elude  altogether  direct  measurement  in 
terms  of  amount.  How  many  thefts  equal  in  wickedness  a  murder  ? 
If  the  piety  of  John  Wesley  is  100,  how  much  is  the  piety  of  St. 
Augustine  ?  How  much  more  ability  as  a  dramatist  had  Shakespeare 
than  Middleton  ?  What  per  cent,  must  be  added  to  the  political 
ability  of  the  Jewish  race  to  make  it  equal  to  that  of  the  Irish  race  ? 
In  these  and  similar  cases  the  quality  to  be  measured  manifests  itself 
objectively  in  so  complicated  and  subtle  effects  that  the  task  of  ex- 
pressing it  in  units  of  amount  is  hopeless. 

Nevertheless,  such  phenomena  can  be  measured  and  subjected  to 


UNITS  OF  MEASUREMENT  19 

exact  quantitative  treatment.  Though  we  cannot  equate  crimes,  we 
can  arrange  them  in  a  list  according  to  their  magnitude,  and  measure 
any  one  by  its  position  in  the  list.  Similarly  St.  Augustine,  if  placed 
in  his  proper  rank  amongst  men  for  piety,  is  measured  as  exactly  as 
if  given  a  numerical  score.  The  step  from  Shakespeare  to  Middle- 
ton  in  a  series  of  dramatists  ranked  in  order  of  ability  is  a  definite 
measure.  If  a  boy  moves  in  English  composition  from  the  position 
of  the  500th  in  a  thousand  to  the  position  of  the  74th  in  a  thousand 
his  gain  is  measured  as  clearly  and  exactly  as  when  we  measure  the 
inches  he  has  grown  in  height.  Measurement  by  relative  position  in 
a  series  gives  as  true,  and  may  give  as  exact,  a  means  of  measurement 
as  that  by  units  of  amount. 

Measurement  by  relative  position  in  scientific  studies  is  of  course 
but  an  outgrowth  of  the  common  practice  of  mankind.  The  man  in 
the  street  measures  things  not  only  as  being  so  many  times  this,  but 
also  as  being  '  the  biggest  he  ever  saw '  or  '  about  average  size.' 

Measures  by  amount  of  some  unit  have  been  the  subject  of  great 
development  in  the  hands  of  physical  science,  while  measures  by 
relative  position  have  been  comparatively  neglected,  though  for  the 
mental  sciences  they  are  of  the  utmost  importance.  The  use  that  has 
been  made  of  them  already  by  Galton,  Cattell  and  others  gives 
promise  that  the  value  of  a  measure  to  which  the  most  subtle  and 
the  most  complex  traits  alike  are  amenable  will  in  the  future  be  more 
appreciated. 

In  measuring  any  person  or  trait  by  position  in  a  series,  the 
chief  desiderata  are  : 

1.  That  the  arrangement  of  the  series  should  not  be  the  result  of 
any  individual's  chance  bias,  i.  e.,  that  the  arrangement  should  rep- 
resent the  general  tendency  of  a  number  of  observers. 

2.  That  it  should  not  be  influenced  by  a  constant  error,  by  bias 
common  to  all,  i.  e.}  that  there  should  be,  on  the  whole,  as  much  bias 
in  any  one  direction  as  in  any  other. 

3.  That  it  should  be  on  a  sufficiently  minute  scale. 

Suppose,  for  instance,  that  we  wish  to  find  the  position  of  a  cer- 
tain theme  among  1,000  English  themes  written  by  first-year  high- 
school  boys.  No  one  person  can,  except  by  accident,  be  a  perfect 
rater  of  these,  for  his  momentary  impulse  or  his  peculiar  ideals  or 
training  will  overweight  certain  features.     The  combined  opinion  of 


20  MENTAL   AND  social   MEASUREMENTS 

ten  equally  good  judges  will  always  be  truer  than  the  opinion  of  any 
one  of  them.  If,  however,  all  the  ten  over-emphasized  spelling  or 
punctuation  or  humor,  their  combined  rating  would  be  false.  Such 
a  oonstanl  error  in  judgment  is  avoided  as  far  as  possible  if  judges  are 
chosen  at  random.* 

The  value  of  having  the  themes  arranged  on  a  fine  scale  is ;  first, 
that  the  finer  the  scale  the  more  precise  the  measure,  and,  second, 
that  if  a  theme  is  then  misplaced  by  chance  it  will  not  be  displaced 
so  far.  For  instance,  if  themes  were  rated  simply  Good  or  Bad,  a 
theme  near  the  dividing  line,  if  put  on  the  wrong  side,  would  be  put 
very  far  to  the  wrong  side,  viz.,  one  fourth  of  the  total  distance, 
whereas  if  they  were  rated  in  20  divisions,  one  in  the  middle  would, 
if  put  to  the  wrong  side,  be  moved  only  one  fortieth  of  the  total  dis- 
tance. As  a  practical  rule  one  should  divide  the  series  into  as  many 
groups  as  one  can  distinguish. 

Amongst  school  abilities,  achievements  in  handwriting,  drawing, 
painting,  writing  English,  translation,  knowledge  of  history,  geogra- 
phy, etc.,  are  readily  measured  by  serial  rating,  and  the  agreement 
of  observers  is  such  that  great  reliance  can  be  put  upon  the  results. 
In  the  case  of  more  general  characteristics  the  service  of  the  method 
will  be  greater  still,  though  the  readiness  and  accuracy  of  the  process 
are  less. 

Measures  by  relative  position  have  one  grave  defect.  Ordinary 
arithmetic  does  not  apply  to  them.  It  is  not  possible  to  add  '  17th 
from  top  of  1,000  in  wealth  '  to  '  92d  from  top  of  1,000 '  as  we  can 
add  fortune  of  $1,000,000  to  fortune  of  $790,000.  We  can  not  say 
that  the  10th  ability  from  the  top  in  100  plus  the  20th  ability  from 
the  top  in  100  is  equal  to  the  14th  plus  the  16th.  We  can  not 
equate  different  positions  in  the  series  with  each  other  as  we  can  dif- 
ferent amounts  of  the  same  thing. 

We  can  not,  that  is,  on  the  basis  of  what  has  been  so  far  said 

about  measurement  by  relative  position  in  a  series.     There  are,  how- 

*Of  course  the  constant  errors  due  to  the  Zeitgeist,  the  general  bias  of  the 
opinion  of  experts  at  any  time,  can  be  overcome  only  by  getting  ratings  made  fifty 
years  apart  !  And  it  is  always  possible  for  the  critic  to  say  that  the  human  judg- 
ments which  we  are  invoking  here,  even  if  the  best  of  their  kind,  are  fallible  ;  that 
the  future  or  Deity  might  in  perfect  wisdom  rate  otherwise  !  This  is  true  enough, 
but  for  the  humble  statistician  the  best  human  judgment  is  all  that  is  needed.  And 
commonly  the  critic's  complaint  that  the  ultimate  structure  of  the  universe  contra- 
dicts a  given  human  judgment  really  means  that  he  himself  does  not  agree  with  it. 


UNITS  OF  MEASUREMENT  21 

ever,  two  possibly  valid  ways  of  transmuting  a  measure  in  terms  of 
relative  position  into  terms  of  units  of  amount.  Given  a  certain  con- 
dition of  the  series  as  a  whole,  and  the  statements  of  position  can  be 
expressed  in  terms  of  amount  and  made  amendable  to  ordinary 
arithmetic.  Given  the  truth  of  a  certain  theory  of  the  amount  of 
difference  noticeable,  and  the  same  result  will  hold.  These  possi- 
bilities will  be  discussed  in  a  special  chapter  on  the  measurement  of 
mental  traits  by  relative  position. 

Problems. 

1.  Why  would  the  number  of  men  giving  instruction  in  a  uni- 
versity not  be  a  fair  measure  of  the  amount  of  teaching  done? 

2.  What  are  the  faults  of  the  following  proposed  as  a  measure 

,,..,.      .         Birth-rate 

ot  civilization  :  ~ — -, —  ? 

Death-rate 

3.  How  could  you  get  commensurate  units  of  amount  of  ability 
in  addition  ?  In  what  sense  could  you,  after  obtaining  such  units, 
say  that  A's  ability  in  addition  was  twice  or  three  times  _B's  ? 

4.  In  giving  examination  marks,  the  custom  is  to  measure  down- 
ward from  a  standard  of  perfection.  Suggest  a  better  starting  point 
to  take. 

5.  What  are  some  objective  units  of  amount  used  to  measure 
criminality  ?  What  would  be  the  advantages  of  measuring  here  by 
relative  position? 

6.  Group  the  following  measures  by  whole  numbers,  first,  by 

using  the  whole  numbers    14,  15,  etc.,  to  represent  13.5-14.499, 

14.5-15.499,  etc.,  and  second  by  using  14,   15,  etc.,  to  represent 

14-14.999,  15-15.999,  etc.: 

18.642,   17.39,  21.45,  14.81,  15.51,  17.23,   19.60,  18.42,  21.7, 
15.861,   16.5,     17.92,  14.4,     19.38,  20.6,     20.5,     18.39,   17.489. 

Which  method  would  you  expect  to  be  the  easier  and  least  subject 

to  error  if  one  had  equal  amounts  of  practice  with  both  ?     Why  ? 

7.  What  is  the  average  salary  of  the  group  represented  by  the 

following  statistics  ?  : 

8  individuals  have  salaries  above  $1,000  and  under  $1,100 
20  "  "  "  ' 

20  "  "  "  ' 

16  "  "  "  ' 

13  "  "  "  ' 

Q  tt  It  tt  t 

g  tl  tt  it  t 


1,100     " 

"         1,200 

1,200     " 

"         1,300 

1,300     " 

1,400 

1,400     " 

"         1,500 

1,500     " 

"         1,600 

1,600     " 

"         1,700 

CHAPTER  III. 

THE    MEASUREMENT    OF    AN    INDIVIDUAL. 

Any  mental  trait  in  any  individual  is  a  variable  quantity.  If 
we  measure  it  a  number  of  times  with  a  fine  enough  scale  of  measure- 
ment we  get  not  one  constant  result,  but  many  differing  results. 
The  amount  of  addition  John  Smith  can  do  in  a  minute,  the  num- 
ber of  cubic  feet  of  sand  Tom  Jones  can  dig  in  an  hour,  the  food 
consumed  by  Richard  Brown  in  a  day,  the  weekly  earnings  of  a  par- 
ticular factory  —  these  and  all  facts  depending  on  human  mental 
traits  are  variable. 

A  constant  can  be  measured  in  a  single  figure,  but  a  variable  for 
its  complete  measurement  requires  as  many  different  figures  as  there 
are  varieties  of  the  thing.  Since  John  Smith  can  add  now  20,  now 
21,  now  22,  now  23  digits  in  a  minute,  his  ability  is  not  any  one  of 
these  nor  the  average  of  them  all,  but  is  described  truly  only  as  20 
such  and  such  a  per  cent,  of  the  times,  21  such  and  such  a  per  cent, 
of  the  times,  etc.  Any  single  figure  would  be  but  an  extremely 
inadequate  representation  of  his  ability  in  addition  or  of  that  of  any 
variable  trait.  The  measure  of  a  variable  quantity  implies  a  list 
of  the  different  quantities  appearing,  with  a  statement  of  the  num- 
ber of  times  that  each  appeared.  Such  a  list  and  statement 
together  are  called  a  table  of  frequencies  or  a  distribution  of  a  trait. 
The  measure  of  a  variable  trait  is  thus  its  entire  distribution 
or  table  of  frequency.  It  is  common  to  present  a  table  of  fre- 
quencies in  a  diagram  in  which  distances  along  a  line  represent 
the  different  quantities,  and  the  heights  of  columns  erected  along  it 
their  frequencies.  Thus  Figs.  1,  2  and  3  represent  at  once  to 
the  eye  the  facts  given  by  Tables  VI.  to  VIII.  Such  a  figure  is 
called  a  surface  of  frequency ;  the  compound  line  which,  with  the 
horizontal  base  line,  encloses  it  is  called  a  distribution  curve. 

Another  method  of  presenting  graphically  a  table  of  frequencies 
is  to  draw  instead  of  the  top  lines  of  the  columns  a  line  joining  the 
middle  points  of  these  top  lines.  Figures  1a,  2a  and  3a  repeat 
Figs.  1,  2  and  3  in  this  form. 

22 


MEASUREMENT  OF  AN  INDIVIDUAL 


23 


1a 


cz 


_o 


3a 


Fig.  1.  —  Surface  of  frequency  of  the  ability  of  B.  F.  A.  in  memory  span.  Num- 
ber of  letters  correctly  written  and  correctly  placed,  after  one  hearing  of  a  series  of 
12.     Number  of  measurements  =  40. 

Fig.  2. — Surface  of  frequency  of  the  ability  of  E.  H.  in  discrimination  of 
length.  Number  of  millimeters  error  made  in  drawing  a  line  to  equal  a  100-mm. 
line.     Number  of  measurements  =  100. 

Fig.  3.  — Surface  of  frequency  of  the  opportunity  for  work  in  a  trade.  Num- 
ber of  members  of  the  Amalgamated  Society  of  Engineers  lacking  employment. 
Number  of  measurements  =31  (years). 

Fig.  1a.  —  Same  as  1,  but  drawn  by  joining  mid-points  of  columns. 

Fig.  2a.  — Same  as  2,  but  drawn  by  joining  mid-points  of  columns. 

Fig.  3a.  — Same  as  3,  but  drawn  by  joining  mid-points  of  columns. 


24 


MENTAL  AM)  social  m  i:asci>i:.mi-:xts 


"            4 

10 

"             3 

7.5 

"             7           ' 

17.5 

"             6 

15 

"             9 

22.5 

"             0 

0 

"             6 

'          15 

"             2 

'            5 

"             1 

2.5 

TABLE   VI. 
Memory  Span  of  B.  F.  A. 
( >f  ;i  Beriee  of  12  letters  read  1  was  correctly  written  and  placed  2  times  or  in    5 


"  "  9 

10 
There  were  40  trials  in  all. 

TABLE   VII. 

Accuracy  of  Discrimination  of  Length  of  E.  H. 

In  drawing  a  line  to  equal  a  100-mm.  line  an  error  of  —    7  mm.  occurred 
<<  a  <(  ii  p. 

It  tl  ((  it  c 

(<  11  l(  11  A 

tt  tt  tl  tl  q 

"  "  "  «  —2 

tt  a  a  a  1 

a  ((  ((  (1  Q 

1 

2 
3 
4 
5 
6 
7 


+ 
+ 
+ 

+ 
+ 
+ 
+ 
+ 
+ 
+  10 


2  times. 
2 


6 
.8 
7 
8 
11 
13 
11 
7 
3 
5 
8 
4 
1 
1 
1 


There  were  100  trials  in  all ;  hence  per  cents.  =  times 

TABLE  VIII.* 
Per  Cent,  per  Year  of  Members  of  the  Amalgamated  Society  of  Engi- 
neers in  Want  of  Employment  During  31  Years. 


s  than  1 

% 

lacked  emplo1 

-ment  in  1  out  of  31 

years,    3. 2 

1 

% 

to    1% 

(< 

8 

tt 

"      25.8 

2 

3 

a 

4 

tl 

"      12.9 

3 

4 

tt 

4 

ii 

"       12.9 

4 

5 

it 

4 

It 

"      12.9 

5 

6 

tt 

•               2 

tt 

6.5 

6 

7 

a 

'              5 

a 

"      16.1 

7 

8 

tt 

'               2 

" 

6.5 

8 

9 

tt 

'               1 

tl 

3.2 

9 

10 

it 

0 

tt 

10 

11 

it 

0 

it 

11 

12 

a 

'               0 

a 

12 

13 

a 

0 

a 

13 

14 

tt 

'           1 

it 

"        3.2 

Tables  IX.-XVL,  and  Figures  4-11  give  each  the  measure- 
ment of  some  variable  trait  in  one  individual. 

*  Arranged  from  data  given  by  George  H.  Wood  on  pages  640-642  of  Vol.  62 
of  the  Journal  of  the  Royal  Statistical  Society. 


MEASUREMENT  OF  AN  INDIVIDUAL. 


25 


TABLE  IX. 
Keaction  Time  H. 


Quantity. 

Frequency. 

Thousandths  of  a  second. 

120-124.9 

9 

125 

18 

130 

35 

135 

37 

140 

43 

145 

36 

150 

38 

155 

40 

160 

38 

165 

18 

170 

24 

175 

11 

180 

15 

185 

20 

190 

10 

195 

3 

200 

4 

205-209.9 

1 

Total  number  of 

measures  taken. 

400 

TABLE  X 

kness  of  Movement  T. 

Quantity. 

Frequency. 

Seconds. 

9.9-10.19 

1 

10.2 

2 

10.5 

6 

10.8 

8 

11.1 

10 

11.4 

18 

11.7 

13 

12.0 

4 

12.3 

4 

12.6-12.89 

1 

67 


TABLE 

XL 

Ability  in 

Addition  S. 

Quantity. 

Frequency, 

Seconds. 

16-16.9 

1 

17 

5 

18 

6 

19 

13 

20 

14 

21 

15 

22 

11 

23 

7 

24 

0 

25-25.9 

2 

Total  number  of 

measures  taken. 

74 

TABLE 

XII. 

ficiency  op  Perception  I 

Quantity. 

Number  of 

letters  seen 

and  marked. 

Frequency. 

6 

2 

7 

5 

8 

15 

9 

20 

10 

24 

11 

16 

12 

5 

13 

0 

14 

1 

26 


MENTAL   AND  social    MEASUREMENTS. 


i_n 


n  n 


120  160  200  0  8  16  24 

4  8 


99  111  123  0  4  8  12 

5  9 


n 


16  21  26  17  21  25  29 

6  10 


CL 


26 


6  10  14  21 

7  11 

I  I*.-.  1  to  11  represent  the  measurements  of  Tallies  IX.  to  XVI.  in  order.     Fig 
4  corresponds  to  Table  IX.,  Fig.  5  to  TaMe  X.,  etc. 


MEASUREMENT  OF  AN  INDIVIDUAL. 


27 


TABLE 

XIII. 

Condition 

of  A  Trade.  * 

Quantity. 

#  out  of 

employment, 

by  years. 

0-1.99 

Frequency. 

Years. 
2 

2.0 

7 

4.0 

6 

6.0 

4 

8.0 

3 

10.0 

3 

12.0 

2 

14.0 

3 

16.0 

18.0 

1 

20.0 

22.0-22.99 

1 

TABLE 

XIV. 

Attendance  of  a  School. 

Quantity. 

Number  absent 

out  of  139 

pupils. 

0  and  1 

Frequency 

Days. 
1 

2    "    3 

19 

4    "    5 

22 

6     "    7 

16 

8     "    9 

7 

10    "  11 

6 

12    "  13 

3 

Total  number  of  measures  taken  32 


74 


TABLE 

XV. 

Exchanges 

of  a  Clearing  P 

Quantity. 

Frequency 

$10,000,000  s. 

17  to  19 

1 

19  "  21 

1 

21  "  23 

7 

23  "  25 

7 

25  "  27 

1 

27  "  29 

2 

TABLE 

VIX. 

Pulse 

of  B. 

Quantity. 

Frequency 

Time  taken 

for  30  beats. 

In  seconds. 

21 

1 

22 

4 

23 

4 

24 

9 

25 

12 

26 

6 

27 

2 

28 

2 

29 

1 

30 

2 

31 

1 

Total  nnmber  of  measures  taken   19 


44 


If  it  were  necessary  to  pick  some  one  kind  of  distribution  as  the 
best  representative  of  all  these,  one  would  choose  that  approached  by 
Figs.  1,  2,  5,  6,  7.  In  them  we  see  the  separate  measures  distrib- 
uted symmetrically  about  a  single  central  measure,  and  decreasing 
in  frequency  as  we  pass  from  the  central  measure  toward  either  ex- 
treme, slowly   at   first,  then    more   rapidly   and   then   more   slowly. 

*  Friendly  Society  of  Iron-founders'  report,  arranged  from  data  given  by  Gr.  II. 
Wood,  Journal  of  the  Royal  Statistical  Society,  Vol.  62,  pp.  640-642. 


28 


MENTAL    AND  SOCIAL    MEASUREMENTS. 


Tln\  follow  roughly  the  type  shown  in  Fig,  12.  But  obviously 
there  i-  do  one  1^ i 1 1« I  thai  adequately  represents  all.  The  number  <>r 
central  typ<  need  nol  be  one,  and  the  variations  from  the  oentral  type 
ni;i\  occur  iii  .-ill  Borts  <>l  ways. 


Fig.  12,  Type  of  distribution  to  which  variable  traits  in  Lndividuala  often 
roughly  approximate<  The  dure  forma  represent  the  same  type  of  distribution}  the 
only  difference  being  in  the  variability, 


Indeed,  even    ill    llir  s:iine  trait    llicre   IliMV  ocelli-  amODg  (lillcniil 

individuals  different  types  of  distribution.  Table  XVII.  and  Fig. 
13  illustrate  this  in  the  case  of  the  aoouraoy  <»C  b  certain  kind 
<>l  perceptive  prooess  in  eleven  individuals.  The  individuals  were 
chosen  at  random  and  so  give  an  impartial  representation  of 
the  fact. 


MEASUREMENT  OF  AN  INDIVIDUAL. 


29 


TABLE  XVII. 
Drawing  a  Link  K^uai,  to  a  100-mm.  Link  Seen. 


Quantity 
Lengl  ii. 

F 

requencies  in  tho 

case 

Of  11  in 

li  vidua]  s. 

71/ 

S 

F, 

0 

£, 

22, 

// 

^ 

A 

2?a 

J? 

81 

1 

2 

3 

1 

4 

5 

4 

6 

2 

7 

3 

8 

1 

9 

1 

90 

1 

2 

1 

2 

1 

1 

2 

3 

3 

1 

1 

2 

1 

4 

4 

2 

2 

5 

1 

6 

1 

3 

6 

11 

6 

9 

7 

3 

8 

3 

7 

7 

0.5 

8 

6 

2 

7 

2 

4 

3 

8 

2 

8 

4 

10 

12 

2 

8 

4 

5 

2 

9 

2 

12 

3 

9 

39 

6 

11 

7 

4 

3 

3 

100 

2 

12 

8 

23 

38 

0 

13 

25 

4 

16 

9 

1 

9 

14 

8 

9 

9 

4 

11 

31 

7 

8 

1 

2 

7 

5 

9 

9 

2 

9 

7 

24 

10 

11 

12 

3 

5 

8 

13 

12 

7 

3 

11 

12 

8 

9 

4 

9 

3 

13 

1 

8 

5 

2 

13 

3 

12 

5 

8.5 

7 

9 

2 

6 

8 

9 

7 

18 

6 

7.5 

G 

2 

(i 

4 

14 

G 

12 

7 

10.5 

5 

4 

1 

7 

5 

4 

8 

8.5 

2 

5 

1 

2 

0 

5 

9 

5 

1 

1 

5 

1 

1 

2 

4 

110 

5 

2 

3 

2 

4 

4 

3 

1 

4 

1 

0 

2 
3 

5.5 

3.5 

3 

2 

1 

1 

1 
1 

4 

2.5 

3 

5 

0.5 

3 

(I 

0 

0 

7 

0.5 

2 

S 

0.5 

0 

9 

1 

120 

0 

1 

0 

2 

1 

3 

0 

4 

1 

30 


MENTAL    AM>   social    MKASCRKMKS'IS. 


N 


n    n 


92 


118      81 


P 


nn 


n  n 


92 


118 


n   n 


ULn 


C   I 


90 


^    n 


III 


IK 


J 


/l 


109 


99         104 


112 


98       102 


-  r  u 


RJ 


Fig.  13.— The  surfaces  of  frequency  that  correspond  to  the  tables  of  frequency  of 
Table  XVII. 


MEASUREMENT  OF  AN  INDIVIDUAL. 


31 


Before  discussing  further  the  treatment  of  a  measure  expressed 
in  a  table  of  frequencies,  it  will  be  well  to  examine  some  clearer  cases 
of  a  hypothetical  nature.  Suppose,  for  example,  that  measures  were 
at  hand  :  (1)  of  the  daily  consumption  of  wealth  by  an  individual, 
(2a)  of  the  hours  worked  daily  by  an  earnest  laborer,  whose  union  did 
not  permit  more  than  an  eight-hour  day,  (26)  of  the  rate  of  addiug 
of  a  practiced  accountant,  (3a)  of  the  amount  of  alcohol  imbibed  daily 
by  a  dipsomaniac,  and  (36)  of  daily  arrests  for  drunkenness  in  a  city. 


B 


Fig.  14. 


An  individual  who  most  frequently  consumes  two  dollars'  worth 
in  food  eaten,  clothes  worn  out,  minor  luxuries,  etc.,  may  consume 
five  dollars'  worth  by  an  expensive  dinner,  ten  dollars'  worth  by 
burning  up  his  coat,  or  a  hundred  dollars'  worth  by  breaking  a  vase 
or  overdriving  a  horse.  He  can  not  consume  less  than  zero.  The 
range  of  distribution  limited  below,  runs  out  above  a  long  way  for 
practically  every  one.  Its  form  will  be  that  of  Type  A  in  Fig. 
14,  a  form  skewed  toward  the  high  end. 

The  laborer  can  not  work  over  eight  hours,  but  will  less  and  less 
readily  suffer  a  greater  and  greater  decrease  from  that  amount  due  to 
weather,  employer's  convenience,  etc.     The  frequency  of  seven-hour 


MENTAL   AND  SOCIAL  MEASUREMENTS. 

days  will  be  much  below  that  of  eight;  that  of  six-hour  days  below 
that  of  seven,  etc  I  omit  from  consideration  Sundays  and  holidays. 
The  form  of  distribution  will  be  that  of  Type  B  in  Fig.  14,  being 
skewed  toward  the  low  end.  So  also  the  practiced  accountant  will 
work  in  most  cases  near  his  best  rate ;  but  while  nothing  can  raise 
him  far  above  his  customary  rate,  distraction  of  attention  by  outside 
stimuli,  fatigue  or  bewilderment  may  drag  him  far  below  it. 

The  periodic  dipsomaniac  drinks  either  a  great  deal  or  little  or 
none,  according  to  the  presence  or  absence  of  the  fit  of  craving.  The 
distribution  of  the  daily  amount  of  liquor  drunk  by  him  will  there- 
fore have  two  points  of  great  frequency,  with  very  slight  frequencies 
for  intermediate  points,  as  shown  in  Type  C  of  Fig.  14.  The 
city's  daily  arrests  for  drunkenness  will  show  a  similar,  though  not 
so  pronounced,  composition  of  great  numbers  due  to  Saturdays,  Sun- 
days and  holidays,  and  smaller  numbers  due  to  ordinary  days.  See 
Type  D  in  Figure  14. 

These  hypothetical  cases  emphasize  types  of  clear  departure  from 
the  common  bell-shaped  form,  and  illustrate  the  insecurity  of  any 
answer  to  our  next  question,  viz.,  How  can  the  main  meaning  of  a 
table  of  frequencies  be  expressed  in  one  or  two  single  figures  capable  of 
treatment  by  ordinary  arithmetic,  or  in  some  simple  algebraic  equation  f 

It  is  customary  to  use  for  any  trait  in  an  individual  his  average 
measure,  but  obviously,  though  the  averages  of  A  and  B  in  Table 
XVIII.  and  Fig.  15  are  identical,  their  abilities  are  widely  differ- 
ent, A  being  a  very  constant  performer,  while  B  is  the  reverse. 
Again  the  average  of  the  man's  daily  consumption  of  wealth  figured 
in  14 A  not  only  does  not  distinguish  him  from  some  one  less  given 
to  extreme  prodigality  who  in  general  lives  on  a  higher  material 
plane,  but  also  gives  no  idea  of  his  common  daily  expenses.  So  also 
the  average  performance  of  the  accountant  does  not  tell  what  is  really 
desired,  namely,  what  the  man  can  do  under  proper  conditions. 
With  a  case  like  that  of  the  dipsomaniac  the  average  grossly  misrep- 
resents the  facts  to  all  readers  who  follow  the  common  habit  of  ex- 
pecting an  average  to  approximate  to  the  individual's  typical  per- 
formance. An  average  is  mathematically  only  the  sum  of  a  set  of 
measures  divided  by  their  number.  It  represents  the  typical  meas- 
ure of  the  set  only  when  there  is  but  one  typical  measure  and  when 
the  set  of  measures  are  symmetrically  disposed  about  it.     There  may 


MEASUREMENT  OF  AN  INDIVIDUAL. 


33 


be  and  often  is  more  than  one  type  of  measure  prominent,  and  the 
distribution  may  be  and  often  is  skewed  instead  of  symmetrical. 

It  is  clear  that  in  every  case  there  are  needed  at  least  two  meas- 
ures, one  of  the  general  tendency  or  typical  performance,  or  measure 


J~~l 


Fig.  15. — The  dotted  line  gives  A's  ability,  the  continuous  line  gives  B's. 
(This  [imaginary  case  is  paralleled  by  many  real  instances.  See,  for  .instance,  C 
and  Lx  in  Fig.  13. ) 

about  which  the  individual  measures  cluster,  the  other  of  the  vari- 
ability or  deviations  from  the  type  or  closeness  of  the  clustering.  If 
there^are  two  or  more  distinct  tendencies  or  types  of  performance  for 
an  individual  a  measure  for  each  is  needed.     If  the  deviations  from 


TABLE 

XVIII 

lanti 

ty. 

Frequency. 

For  A. 

For  B 

21 

2 

22 

4 

23 

3 

8 

24 

7 

9 

25 

20 

16 

26 

2K 

18 

27 

22 

15 

28 

7 

11 

29 

1 

7 

30 

5 

31 

1 

the  type  follow  different  gradations  above  and  below  it,  as  in  skewed 
distributions,  separate  measures  are  needed  for  those  above  and  below. 
In  general,  so  far  as  the  frequencies  of  different  degrees  of  deviation 
follow  no  simple  law,  no  single  figure  can  describe  them. 

It  is  customary  to  use  for  a  measure  of  any  mental  trait's  vari- 
ability in  an  individual  the  average  or  mean  deviation  of  the  separate 

3 


34 


MENTAL   A\I>  social   Ml'Asri:HMI-:XTS. 


measures  from  their  average.  But  the  considerations  just  mentioned 
and  the  foot  that  variability  may  be  extremely  irregular  disallow  any 
such  naive  procedure.  The  amount  drunk  by  the  dipsomaniac  in 
the  illustration  really  varies  little,  provided  we  take  him  in 
drinking  tits  alone  or  in  sober  conditions  alone,  but  the  single 
figure  of  the  mean  variation  would  picture  a  man  of  wide  range 
day  by  day. 

The  only  set  of  figures  which  adequately  represent  a  variable 
measure  in  an  individual  are  those  from  which  the  entire  table  of 


Fig.  16. 


Fig.  16. — Distribution  of  a  quantity  with  Average  10;  Average  Deviation 
from  it  2 ;  form  of  the  surface  of  frequency,  a  rectangle. 

Fig.  17.  —  Distribution  of  a  quantity  with  Average  10  ;  Average  Deviation 
from  it  2  ;  form  of  the  surface  of  frequency,  that  of  the  normal  probability  integral. 


frequency  can  be  calculated,  which  present  it  in  briefer  space  and 
more  convenient  manner,  but  unaltered.  In  certain  cases  two  or 
three  figures  with  a  statement  of  the  general  form  of  the  distribution 
could  do  this.  Thus,  "Av.  10.  Average  deviation  2.  Form  of 
distribution,  a  rectangle,"  tells  us  that  the  distribution  is  that  of 
Fig.  16.  So  also  "  Av.  10.  Average  deviation  2.  Form  of  dis- 
tribution, that  of  the  surface  of  frequency  of  the  normal  probability 
integral,"  tells  the  student  who  is  acquainted  with  certain  facts  that 
the  distribution  is  that  of  Fig.  17. 

It  is  obvious  that  if  the  distribution  does  not  take  some  regular 


MEASUREMENT  OF  AN  INDIVIDUAL.  35 

form  it  can  not  be  represented  by  a  simple  algebraic  expression.* 
In  certain  cases,  where  it  does  take  such  a  regular  form,  it  can  be  so 
represented.  Thus  if  a  man's  earnings  ranged  from  A  to  B  per  day 
and  were  one  as  often  as  another  of  these  values,  the  surface  of  fre- 
quency would  be  the  rectangle  with  base  AB,  and  with  height  de- 
termined by  the  number  of  individual  measures  and  the  scale  taken 
for  the  frequencies.  In  algebraic  language,  letting  x  equal  the  quan- 
tity and  y  the  frequency,  y  =  K  or  0,  K  for  values  of  x  between  A 
and  B,  0  for  all  other  values  of  x. 

If  a  man's  daily  earnings  varied  from  A  to  B,  decreasing  in  fre- 
quency in  arithmetical  progression  as  the  amount  increased  until  a,t 
B  the  frequency  was  0,  the  surface  of  frequency  would  be  made  up 
of  such  a  series  of  rectangles  of  equal  base  as  would  be  inscribed  in 
a  right-angled  triangle.  The  rate  of  decrease  would  decide  the  slope 
of  the  triangle's  hypotheneuse.  As  the  amount  of  earnings  was  dis- 
tributed on  a  finer  and  finer  scale  the  surface  of  frequency  would 
more  and  more  approach  a  right-angled  triangle,  the  mode  being  one 
side.  Y  would  equal  K(B  —  x)  within  the  limits  of  x=  A  and 
x  =  B,  and  0  for  all  other  values  of  x.  K  would  be  a  constant 
measuring  the  rate  of  decrease. 

If  the  man's  earnings  varied  from  A  to  B,  the  frequency  increas- 
ing in  arithmetical  progression  from  0  at  A  up  to  C  and  decreasing 
regularly  in  the  same  progression  from  then  on  to  0  at  B,  the  sur- 
face of  frequency  would  approach  as  a  limit,  a  finer  and  finer  scale  of 
amounts  being  used,  an  isosceles  triangle  with  base  AB.  The  slope 
of  its  two  sides  would  be  decided  by  the  rate  of  increase  and  decrease 
as  measured  by  a  constant  K.  Y  would  equal  K  {x  —  A)  for  values 
of  x  from  A  to  C\  K  (B  —  x)  for  values  of  x  from  C  to  B,  and  0  for 
all  other  values  of  x. 

If  a  man's  earnings  on  any  one  day  were  due  to  the  action  of  one 

combination  out  of  all  the  possible  combinations,  all  equally  likely  to 

occur,  of  an  infinite  number  of  causes  equal  in  amount  and  independ- 

*  As  the  scale  of  measurement  is  made  finer  the  top  of  the  surface  will  of  course 
tend  to  become  a  continuous  line.  For  it  then  some  mathematical  expression  can 
be  discovered.  The  relation  of  the  vertical  distance  representing  frequency  to  the 
horizontal  distance  representing  quantity  is,  of  course,  the  relation  actually  shown 
in  the  curve  and  to  be  shown  algebraically.  The  frequency  is  commonly  called  y 
and  the  quantity  x.  Or  if  the  distribution  curve  is  drawn  in  the  manner  shown  mi 
page  23  (by  joining  the  middle  points  of  the  top  of  the  rectangles),  the  inquiry  may 
be  made  as  to  the  expression  which  will  best  satisfy  that  series  of  points. 


36  MENTAL    AXD  SOCIAL    M K.lsl'i; KM KSTS. 

enl  of  each  other,  the  distribution  would   be  of  the  sort  shown   in 
Figures  L2  and    17.     The  equation  would  be  (if  P  =  the  maximum 

ordinate) 

y  =  l\-">"'      or     y\=  c~x" 
or  some  specialized  form  (e.g., 

i     r'\ 

y  =  — 7=e  *', 

in  which  case  p.  gives  a  measure  of  the  variability  of  the  trait).  * 

This  last  case  is  identical  with  the  last  case  of  the  description  of 
a  distribution  by  two  single  figures.  The  surface  of  frequency 
thus  obtained  is  that  to  which  the  bell-shaped  distributions  often 
approximate.  If  it  is  constructed  from  an  infinite  number  of  indi- 
vidual measures,  its  average,  mode  and  median  exactly  coincide. 
They  are  approximately  coincident  when  the  distribution  is  of  only 
a  small  number  of  measures,  the  differences  between  them  being  in 
the  long  run  greater  the  smaller  the  number  of  measures  is.  A  de- 
viation of  any  amount  above  the  average  is  with  an  infinite  number 
of  measures  of  the  same  frequency  as  a  deviation  of  the  same  amount 
below.  It  is  of  approximately  the  same  frequency  when  a  limited 
number  of  measures  are  taken.  The  frequency  of  deviations  de- 
creases with  their  amount,  first  slowly,  then  rapidly  and  then  slowly 
again.  It  is  called  the  curve  of  error  or  the  normal  type  of  distri- 
bution. Its  properties  will  be  more  fully  described  in  Chapters  IV. 
and  V.  The  frequency  with  which  traits  in  an  individual  are  ap- 
proximately so  distributed,  the  nature  of  the  traits  in  such  cases  and 
the  closeness  of  the  approximation,  have  hardly  been  studied. 

Concerning  the  algebraic  expression  of  a  table  of  frequencies,  the 
warning  of  page  34  must  be  repeated  : 

The  only  equation  or  set  of  equations  which  adequately  represent 
a  variable  measure  in  an  individual  are  those  from  which  the  entire 
table  of  frequencies  can  be  calculated,  which  present  it  in  briefer 
space  or  more  convenient  form,  but  unaltered. 

From  all  these  considerations  a  few  simple  rules  emerge  : 
1.  The  real  measure  of  a  variable  trait  in  an  individual  is  the 
table  of  frequencies. 

*The  reader  unfamiliar  with  higher  algebra  will  have  to  take  this  on  faith. 


MEASUREMENT  OF  AN  INDIVIDUAL.  37 

2.  Beware  of  inferring  too  much  from  any  single  measure  or  few 
measures  of  an  individual.  * 

3.  Always  turn  a  series  of  measures  into  a  table  of  frequencies 
before  inferring  anything  from  them. 

4.  Never  replace  a  table  of  frequencies  by  mere  measures  of 
their  average  and  mean  variation  until  simplification  is  necessary. 

5.  Never  write  about  an  average  or  a  mean  variation  without  an 
accurate  description  of  the  type  of  distribution  whence  it  came.  It 
is  probably  wise  to  print  every  distribution  in  detail. 

When  the  distribution  can  be  described  by  two  measures,  one  of 
general  tendency  and  one  of  variability,  and  when  it  is  necessary  to 
use  such  measures  even  though  they  give  only  an  inaccurate  descrip- 
tion, the  following  points  should  be  borne  in  mind  : 

Two  other  measures  of  a  variable  trait,  the  median  and  the  mode, 
are  often  more  serviceable  than  the  average  and  are  commonly  use- 
ful in  addition  to  it. 

The  median  is  the  measure  above  which  and ,  below  which  are 
equal  numbers  of  the  separate  measures. 

The  mode  is  the  most  frequent  measure. 

The  mode  is  especially  helpful  in  the  case  of  distributions  show- 
ing two  or  more  types  of  performance  by  the  same  individual,  for 
each  type  can  be  represented  by  a  different  mode  and  its  relative 
importance  by  its  mode's  frequency. 

The  following  characteristics  of  the  different  measures  may  help 
to  decide  which  is  the  best  to  use  in  any  given  case : 

The  mode  is  the  most  easily  and  quickly  determined.  It  is  not 
so  reliable  a  measure  as  the  others.  That  is,  the  actual  mode  ob- 
tained from  a  given  number  of  cases  will  not  be  so  near  the  true 
mode  as  will  the  actual  average  to  the  true  average.  In  reality, 
however,  since  the  mode  is  commonly  taken  on  a  much  rougher  scale 
than  the  average,  it  is  really  often  just  as  reliable,  only  less  precise. 
It  is  hardly  at  all  influenced  by  extreme  measures  or  erroneous 
measures.  It  is  entirely  unambiguous  and  does  not  mislead  a  reader 
into  thinking  that  all  the  individual  measures  of  a  group  are  very 
closely  near  it. 

The  median  is  more  easily  determined  than  the  average.  It  is 
not  so  precise  as  the  average,  is  very  little  influenced  by  extreme 
or  erroneous  measurements  and  is  unambiguous. 

*  The  number  needed  will  be  discussed  in  Chapter  X. 


MENTAL   AND  SOCIAL   MEASUREMENTS. 

The  average  is  determined  only  with  considerable  arithmetical 
work,  hut  this  same  work  gives  the  variability  as  well.  It  is  more 
precise  than  the  mode  or  the  median  because  the  amount  of  every 
measure  plays  a  part  in  determining  it,  but  for  this  very  reason  it  is 
more  influenced  by  extreme  or  erroneous  measures.  The  average  is 
the  measure  in  common  use  and  has  the  advantage  of  being  a  famil- 
iar term,  and  at  the  same  time  the  disadvantage  of  leading  untrained 
readers  to  think  that  the  abilities  of  which  it  is  the  average  are 
closely  clustered  about  it. 

Measures  of  the  variability  or  closeness  of  clustering  of  the  indi- 
vidual measures  are  of  two  sorts.  There  are  measures  of  the  average 
of  the  deviations  of  the  individual  measures  from  their  central 
measure,  and  measurements  of  the  limits  above  and  below  the  central 
measure  which  include  a  certain  proportion  of  all  the  individual 
measures. 

Of  the  first  sort  wTe  have  the  average  deviation,  wThich  equals  the 
average  of  the  deviations  of  the  individual  measures  from  their  aver- 
age, median  or  mode ;  and  the  mean  square  deviation  or  standard 
deviation,  which  equals  the  square  root  of  the  average  of  the  squares 
of  the  deviations  of  the  individual  measures  from  their  average, 
median  or  mode.  Of  the  second  sort  the  measure  in  common  use  is 
the  probable  error,  or  P.  E.,  which  gives  the  distance  which  must 
be  taken  above  and  below  the  average,  median  or  mode,  in  order  to 
include  between  the  two  limits  thus  obtained  50  per  cent,  of  all  the 
individual  measures.  We  can,  however,  calculate  in  a  similar  way 
the  limits  needed  to  include  10,  20,  75,  90  or  any  other  per  cent,  of 
the  individual  measures,  and  can  reckon  deviations  from  any  point 
as  well  as  from  the  central  tendency,  if  we  choose. 

Strictly  speaking,  measures  of  the  first  sort  are  calculated  only 
from  the  average,  but  it  is  entirely  allowable  to  reckon  them  from 
the  mode  or  median  if  a  statement  is  made  that  this  is  done. 

Measures  of  the  first  class  are  the  more  reliable  in  the  sense  that 
if  the  measures  for  the  separate  trials  are  reliable  the  same  number 
gives  an  average  deviation  or  deviation  of  mean  square  more  exactly 
than  it  gives  the  probable  error.  They  are,  however,  more  influenced 
by  erroneous  or  extreme  measures. 

In  the  case  of  skew  distributions  the  mode  is  in  general  the  most 
advantageous  measure  of  general  tendency ;  the  variabilities  above 
and  below  it  should  be  given  separately. 


MEASUREMENT  OF  AN  INDIVIDUAL.  39 

In  the  case  of  multimodal  distributions  the  different  modes 
should  each  be  stated ;  the  total  table  of  frequencies  should  be 
analyzed  into  different  distributions,  one  for  each  of  the  different 
modes  ;  these  distributions  should  be  treated  separately  by  the  above 
rules. 

The  statement  of  the  limits  needed  to  include  20  to  30  per  cent, 
of  the  cases  is  often  a  convenient  expression  of  typical  performance, 
giving,  as  it  does,  a  wide  mode. 

If  the  measures  of  an  individual  are  not  in  terms  of  amount,  but 
are  simply  a  ranked  series  of  acts  of  kindness,  or  poems,  or  crimes, 
or  examination  papers  in  Latin  or  geography  or  English  themes,  the 
only  measures  of  central  ability  that  we  can  use  are,  of  course,  the 
mode  or  the  median  ;  of  these  the  mode  is  commonly  the  most  in- 
structive. The  only  measures  of  variability  that  can  be  used  are 
measures  by  limits  including  a  given  percentage. 

Finally,  it  is  a  safe  rule  to  ask  concerning  any  figure  derived 
from  a  distribution  of  a  variable  trait,  'Just  what  real  quantity  in 
the  man  does  this  figure  represent  ? '  and  to  use  the  figure  only  when 
a  definite  answer  can  be  given. 

Problems. 

8.  Express  in  tables  of  frequencies  and  surfaces  of  frequency  the 
following  facts  : 

Ar.,  being  measured  with  respect  to  his  memory  span  for  letters  40 
times,  showed  the  following  abilities,  in  terms  of  the  number  of  words 
remembered  in  their  correct  positions :  7,  6,  7,  5,  8,  2,  10,  6,  7,  8, 
3,  8,  6,  9,  6,  10,  6,  8,  6,  4,  9,  6,  10,  8,  6,  8,  5,  6,  4,  8,  10,  7,  4,  7, 
6,  9,  1,  11,  7,  7. 

D.,  being  measured  in  the  same  trait  40  times,  showed  records  of: 
5,  4,  1,  6,  5,  5,  8,  4,  6,  5,  5,  5,  4,  6,  4,  4,  5,  7,  2,  5,  5,  4,  5,  4,  6, 
9,  4,  3,  0,  5,  5,  6,  5,  6,  3,  8,  4,  5,  5,  3. 

9.  Which  is  the  more  variable,  Ar.  or  D.  ? 

10.  What  is  the  average  deviation  of  each  from  his  mode  ? 

11.  In  which  case  is  it  almost  a  matter  of  indifference  whether 
the  general  tendency  is  expressed  by  the  average  or  by  the  median 
or  by  the  mode  ? 

12.  Is  it  a  matter  of  indifference  in  the  case  given  in  ques- 
tion 13? 


10  MENTAL   AND  social   MEASUREMENTS. 

13.  The  percentages  of  workmen  out  of  employment  in  England 
were,  for  different  years  from  1860  to  1891,  0-.99,  no  years;  1-1.99, 
9  years;  2-2.99,  10  years  j  3-3.99,  4  years;  4-4.99,  7  years;  5- 
").!,!i,  1  year.  Comparing  this  table  of  frequencies  with  that  given 
in  Table  XIII.,  which  was  the  worse  on  the  whole,  the  condition 
with  respect  to  getting  employment  of  workmen  in  general  or  that  of 
the  members  of  the  Friendly  Society  of  Iron  Founders?  Which 
was  the  more  variable  ? 

14.  Why  Mould  not  the  average  be  a  sufficient  measure  of  the 
general  tendency  of  an  individual's  body-temperature  ? 

1 5.  What  would  be  the  probable  form  of  distribution  of  the  daily 
traffic  of  a  city's  street-railroad  system  ? 


CHAPTER   IV. 

THE    MEASUREMENT    OF    A    GROUP. 

The  sciences  of  human  nature  commonly  use  measures  of  indi- 
viduals only  in  order  to  get  measures  of  groups.  Not  John  Smith's 
spelling  ability,  but  that  of  all  fifth  grade  boys  taught  by  a  certain 
method  ;  not  A's  delicacy  of  discrimination  of  weight,  but  that  of  all 
men ;  not  B's  wage,  but  that  of  all  railroad  engineers  during  a  cer- 
tain period  ;  not  the  number  of  C's  children,  but  the  productivity  of 
the  English  race  as  a  whole  ;  not  individuals,  but  groups,  are  com- 
monly to  be  measured,  compared  and  argued  about. 

The  customary  expression  of  a  trait  or  ability  in  a  group  is  its 
average,  and  the  use  of  an  average  here,  as  before,  points  to  the 
variability  of  the  fact.  We  do  not  seek  the  average  law  of  gravity, 
or  the  average  ratio  of  amount  of  oxygen  to  amount  of  hydrogen  in 
an  atom  of  water,  or  the  average  velocity  of  sound.  It  is  because  of 
the  unlikeness,  the  variability,  of  even  the  most  similar  human  indi- 
viduals in  even  the  most  constant  human  qualities  that  we  are  forced 
to  use  averages  at  all. 

An  average  no  more  represents  the  different  abilities  of  the  mem- 
bers of  a  group  than  it  did  the  different  measures  of  a  trait  in  a 
single  individual.  The  thing,  trait  A  in  group  X,  is  a  variable 
quantity  and  is  measured  only  by  a  list  of  the  different  degrees  of 
the  trait  found  in  all  the  individuals  of  the  group,  with  a  statement 
of  the  number  of  times  each  appears.  A  table  of  frequencies  or  sur- 
face of  frequency  will  be  the  adequate  measure  here,  as  before.  The 
measure  of  a  trait  in  a  group  is  its  total  distribution,  and  this  total 
distribution  is  simply  all  the  separate  measures  of  the  individuals 
making  up  the  group. 

The  measure  taken  for  each  individual  may  be  his  average  or  his 
most  frequent  ability  or  highest  ability  shown,  or  lowest  ability  shown, 
or  ability  exceeded  in  50  per  cent,  of  his  trials,  or  ability  exceeded 
in  70  per  cent,  of  his  trials,  or  variability,  or  total  distribution,  or  any 
other  characteristic  of  "  individual  in  group  X." 

Most  frequently  some  measure  of  central  tendency  is  the  one  to 

41 


42 


MENTAL    A.XD   SOCIAL    Mi:\sr HVMVXTS. 


be  used.  In  such  cases  the  individual  measures  may  be  from  very 
['«  w  trials  without  doing  much  harm.  In  fact,  an  accurate  repre- 
sentation of  tht'  ability  of  a  group  may  arise  from  very  inaccurate 
measurements  of  the  individuals  iu  it ;  for  instance,  from  measure- 
ments from  onlv  a  single  record  from  each  individual.     The   reason 


L_n 


2 

-4 

+4 

+  12 

18 

1 
1 

i 

:        I 

125  200  275 

19 


^£1 


-22       -12        -2         +8        +18 

20 

Fi'.s.  18,  19  and  20  present  graphically  the  facts  of  Tables  XIX.,  XX.  and 
XXI.  respectively. 

is,  of  course,  that  the  errors  being  chance  errors,  the  too  high  rating 
of  A  is  counterbalanced  by  the  too  low  rating  of  B,  and  so  on,  so 
that  with  hundreds  of  cases  the  central  tendency  of  the  distribution 


MEASUREMENT  OF  A   GROUP.  43 

is  unchanged.  Thus  the  continuous  line  in  Fig.  18  gives  the  dis- 
tribution of  the  averages  of  100  individuals'calculated  from  only  4 
instead  of  from  20  records  from  each,  the  4  being  chosen  at  random 
from  the  20.  The  broken  line  gives  the  distribution  when  all  the 
20  are  used.  Table  XIX.  gives  the  facts  in  figures.  Table  XX. 
and  Fig.  19  give  the  distribution  of  the  cost  per  pupil  of  supplies  in 
40  grammar  schools  for  boys  in  New  York  City  calculated  from  2 
and  from  4  years'  figures  respectively.  It  is  evident  that  one  would 
not  be  much  misled  with  respect  to  the  general  tendency  of  the 
group  by  taking  the  measure  of  the  group  from  4  records  instead  of 
that  from  20,  or  even  that  from  2  instead  of  that  from  4. 

When  the  measure  taken  for  an  individual  is  his  total  ability, 
the  measure  of  the  group  is,  of  course,  a  total  distribution  made  up  of 
all  the  separate  individuals'  distributions,  each  individual  being  given 
his  proper  share  in  determining  this  total  distribution.  In  practice 
we  rarely  make  up  the  total  distribution  of  a  trait  in  a  group  from 
adequate  individual  distributions,  but  use  for  each  individual  only  a 
few  measures.     The  result  is  very  closely  the  same  if  the  number  of 

TABLE  XIX. 

Average  Error  in  drawing  a  Line  to  Equal  a  100-mm.  Line. 
A  =  averages  calculated  from  20  trials  for  each  individual. 
B  =  averages  calculated  from  4  trials. 


Quantity  :  in  tenths 
of  millimeters. 

Frequen 
A. 

cies. 
B. 

Quantity  :  in  tenths 
of  millimeters. 

Frequencies. 
A.             B. 

—  100  to 

—  120 

1 

.  1 

+  40  to  +  60 

5              6 

—  80  to 

—  100 

3 

4 

-4-  60  to  +  80 

4              1 

—  60  to 

—  80 

7 

5 

+  80  to  +  100 

3            3 

—  40  to 

—  60 

12 

5 

+  100  to  +  120 

0 

—  20  to 

—  40 

17 

18 

+  120  to  +  140 

1 

0  to 

—  20 

18 

24 

Averages      — .  72  mm 

.     — .46  mm. 

0  to 

+  20 

13 

17 

Medians        — .  889  mm.  —  .584  mrr 

+  20  to  4-  40 

17 

15 

TABLE 

XX. 

Cost  per 

Pupil  op 

General  Supplies  in  40  Boys'  Grammar  Schools. 

Quantity,        Frequency, 
Dollars.      4  records  used. 

Frequency, 
2  records  used. 

Quantity,        Frequency, 
Dollars.     4  records  used. 

Frequency, 
2  records  used. 

1.25-1.50 

1 

2 

2.75                   7 

6 

1.50-1.75 

1 

0 

3.00                   3 

5 

1.75,  etc. 

2 

5 

3.25                  1 

1 

2.00 

6 

7 

3.50 

1 

2.25 

13 

5 

Averages        2.48 

2.49 

2.50 

6 

8 

Medians          2.44 

2.53 

44  MENTAL   ASP  SOCIAL   MEASUBEMENTS. 

individuals  is  large.  Thus  the  broken  ami  continuous  lines  of  Fig. 
20  -hew  practically  the  same  fact,  though  the  former  gives  the 
measure  of  the  total  ability  of  the  group  made  up  by  putting  together 
all  the  separate  distributions  from  20  trials  each,  while  the  latter 
gives  the  total  ability  made  up  by  putting  together  only  4  from  each. 
Table  XXI.  gives  the  facts  in  figures. 

TABLE  XXI. 

Errors   Made  r.v  92  Indiviiu'als   in  Drawing  a  Line  to  Equal  a  IOO-mm. 

Lini:.     .1  Gives  the  Distribution  Due  to  20  Trials  from  Each 

I.vi>!\ 'U'Ual,  £  that  Due  to  4.*    B  is  Raised  to  an 

Equivalence  to  Make  Comparison  Easier. 


Quantity. 

Error  from  standard, 

in  nuns. 

Frequencies. 
A.               B. 

Quantity. 
Error  from  standard, 
in  mms. 

Frequencies. 
A.              B. 

16  or  less 

9 

10 

+ 

3 

187 

177 

14 

13 

0 

+ 

5 

103 

US 

12 

21 

25 

+ 

7 

89 

108 

10 

62 

64 

9 

47 

44 

8 

84 

94 

11 

33 

39 

—   6 

107 

123 

13 

13 

10 

—   4 

217 

207 

15 

6 

10 

o 

262 

197 

17 

4 

0 

0 

292 

306 

19  and  over 

H 

10 

+    1 

256 

237 

The  determinations  of  the  central  tendency  and  variability  of  a 
measure  of  a  group  are  made  in  just  the  same  way  as  in  the  case  of  a 
measure  of  an  individual,  and  the  different  measures  of  them  have 
here  the  same  characteristics.  The  formal  and  mathematical  problem 
is  identical  whether  we  have  varying  records  of  one  individual  or 
varying  individuals  of  one  group,  or  varying  records  of  many  indi- 
viduals in  one  group. 

Starting,  then,  with  the  best  measures  of  the  individuals  (for  our 
purpose)  that  can  be  obtained,  we  put  them  together  in  a  total  distri- 
bution (allowing  equal  weight  to  each)  and  have  the  measure  of  the 
group.  As  in  the  case  of  individual  measures,  it  is  a  safe  rule  never 
to  replace  this  totality  by  any  partial  expressions  of  it  until  it  is 
necessary.  As  in  the  case  of  an  individual  measured,  the  distri- 
butions may  conceivably  take  all  sorts  of  forms  and  be  quite  unrep- 
resentable by  any  simple  arithmetical  constants. 

But  in  point  of  fact  the  measurements  of  groups  with  which 

*  There  were  less  than  20  trials  in  a  few  cases,  hence  the  total  numbers  are  not 
exactly  1840. 


MEASUREMENT  OF  A    GROUP.  45 

students  of  mental  science  have  to  deal  do,  in  the  case  of  most  ana- 
tomical traits,  of  very  many  physiological  traits,  of  many  mental 
traits  and  of  at  least  some  institutional  and  social  traits,  show  an 
approximation  toward  a  distribution  the  variability  of  which  is  of 
such  a  nature  as  to  justify  one  in  regarding  the  members  of  the  group 
as  representatives  clustering  about  a  type,  departures  from  which  show 
a  certain  regularity.  In  other  words,  the  statistical  average  or  mode 
very  often  represents  a  real  central  tendency  or  type,  and,  the  de- 
partures from  it  occurring  in  an  orderly  way,  one  or  two  figures  can 
often  represent  the  real  clustering  of  individuals  about  a  type. 

In  particular  there  is  found  very  often  a  form  of  distribution  ( 1 ) 
approximating  the  symmetrical,  with  its  mode  approximately  at  the 
average,  so  that  both  are  nearly  coincident  with  the  median,  and  (2) 
characterized  by  a  slow  decrease  in  frequency  for  a  certain  distance 
above  and  below  the  mode,  a  more  rapid  decrease  from  then  on  for  a 
way,  and  finally  a  slower  decrease  until  the  limits  are  reached.  This 
description  the  reader  will  recognize  as  the  description  of  a  distribu- 
tion approximating  to  the  so-called  normal  distribution,  that  of  a  quan- 
tity determined  by  the  action  of  a  large  number  of  independent  causes 
equal  in  amount ;  in  other  words,  that  of  the  probability  curve. 

In  so  far  as  this  uniformity  in  distributions  does  exist,  we  are 
freed  from  the  necessity  of  devising  a  separate  means  of  quantitative 
expression  for  each  group  measurement  studied,  and  permitted  to 
express  it  at  least  approximately  in  two  figures,  one  telling  the 
general  tendency  or  type,  the  other  the  variability.  The  average, 
median  and  mode  as  measures  of  general  tendency,  and  the  average 
deviation,  standard  deviation,  P.  E.,  etc.,  as  measures  of  variability, 
possess  perhaps  a  wider  and  surer  utility  in  the  case  of  measures  of 
groups  than  in  the  case  of  measures  of  individuals.  The  properties 
of  the  probability  curve  become  of  practical  importance. 

I  have  represented  graphically  in  the  following  pages  distribu- 
tions of  as  many  anatomical,  physiological,  mental,  social  and  insti- 
tutional traits  as  I  could  conveniently  collect,  drawing  them  so  that 
a  rough  comparison  with  the  surface  of  frequency  of  the  probability 
integral  could  be  made  in  each  case.*     The  examination  of  these  will 

*  The  author  will  be  much  indebted  to  any  of  his  readers  who  sends  him  the 
table  of  frequencies  of  any  trait  measured  in  any  group,  especially  if  the  group  is  a 
large  one.  Such  data  must  be  at  hand  in  any  large  hospital,  school,  psychological 
laboratory  or  gymnasium. 


46  MENTAL   AND  social   MEASUREMENTS. 

give  a  concrete  and  reasonably  accurate  notion  of  the  frequency  with 
which  the  measurement  of  a  group  is  again  and  again  approximately 
the  same  statistical  problem. 

In  these  figures  f-1  to  47)  the  continuous  lines  enclose  the  sur- 
face of  frequency  of  the  trait  in  question.  The  dotted  lines  give  the 
surface  which  would  be  found  if  the  distribution  of  the  trait  followed 
the  type  of  the  normal  distribution,  the  probability  surface.  Where 
the  actual  distribution  obviously  does  not  follow  this  type  even 
approximately,  the  dotted  lines  are  omitted.  The  exact  nature  of 
the  trait,  the  number  of  individuals  and  the  source  of  the  data  in 
each  ease  are  given  in  the  list  that  follows.  When  no  source  is 
stated  the  author  is  responsible  for  the  original  data. 

Fig.  21. — Height  of  American  adult  men.  In  inches.  V  (number  of  cases) 
=  25,878.  Drawn  from  the  table  given  by  Karl  Pearson  on  page  385  of  Vol.  186A 
of  the  Philosophical  Transactions  of  the  Royal  Society  of  London.  He  quotes  from  J. 
H.  Baxter,  Medical  Statistics  of  the  Provost  Marshal's  General  Bureau. 

Fig.  22. — Weight  of  English  adult  men.  In  pounds.  jV  =  5,552.  Drawn  from 
the  table  given  in  C.  Roberts'  '  Manual  of  Anthropometry '  ;  appendix. 

Fig.  23.  —  Cephalic  Index  (ratio  of  width  to  length  of  head)  of  modern  Alt- 
Baverische  skulls.  JV=900.  Drawn  from  the  table  given  by  Karl  Pearson  in 
'  The  Chances  of  Death.' 

Fig.  24.  —  Length  of  male  infants  at  birth.  In  inches.  AT=451.  Source  the 
same  as  for  Fig.  22. 

Fig.  25.  —  Girth  of  chest,  empty,  of  English  army  recruits.  In  inches.  N= 
675.     Source  the  same  as  for  Fig.  22. 

Fig.  26.  —  Strength  of  arm  pull.  English  adult  men.  Pull  exerted  as  in  draw- 
ing a  bow.     In  pounds.     N=  1497.     Source  the  same  as  for  Fig.  22. 

Fig.  27.  —  Body  temperature  at  the  mouth  in  American  women.  N=  158.  I 
am  indebted  for  the  original  measures  to  Professor  T.  D.  Wood,  of  Teachers  College. 

Fig.  28.  —  Heart  rate  (after  vigorous  exercise)  in  American  students,  young 
men  16  to  20.  Number  of  beats  per  60  seconds.  X=  312.  I  am  indebted  for  the 
original  measures  to  Dr.  G.  L.  Meylan,  of  Columbia  University. 

Fig.  29.  — Beaction  time  of  American  college  freshmen.  Thousandths  of  a  sec- 
ond. N=252.  lam  indebted  for  the  original  measures  to  Dr.  Clark  Wissler,  of 
the  American  Museum  of  Natural  History. 

Fig.  30  —  Memory  span  for  digits  in  American  women  students.  Number  of 
digits  correctly  written  and  correctly  placed.     N=  123. 

Fig.  31.  —  Efficiency  in  perception  of  12.5-year-old  boys.  Number  of  A's 
marked  in  60  seconds  on  a  sheet  of  13  lines  of  capital  letters  (see  sample  below). 
#=312. 

OYKFIUDBHT A<  I  DA ACDIXAMRPAGQZTAACV  AOWLYXWABBTHJJANE 
EFAAMEAACBSVSKALLPHANRNPKAZFYRQAQEAXJUDFOIMWZSAUC 
(rVAOABMAYDYAAZJDALJACTNEVBGAOFHABPVEJCTQZAPJLEIQWN 
AHRBULAS 

Fig.  32. — Efficiency  in  controlled  association  of  12.5-year-olds.  Number  of 
correct  minus  number  of  incorrect  opposites  of  the  following  words  written  in  60 


MEASUREMENT  OF  A   GROUP.  47 

seconds:  Good,  outside,  quick,  tall,  big,  loud,  white,  light,  happy,  false,  like,  rich, 
sick,  glad,  thin,  empty,  war,  many,  above,  friend.     iV=239. 

Fig.  33.  — Accuracy  of  estimation  of  length  in  girls  13  to  15  years  old.*  Aver- 
age variable  error,  in  millimeters,  in  30  attempts  to  draw  a  line  equal  to  a  100-mm. 
line  seen.     N=  153. 

Fig.  34.  — Efficiency  in  complex  perception  of  12.5-year-old  boys.  Number  of 
words  containg  a  and  t  marked  in  120  seconds  in  a  sheet  of  words  (see  sample  be- 
low).    iV=312. 

Dire  tengo  antipatia  senores  ;  esto  seria  necedad,  porque  hombre  vale  siempre 
tanto  como  otro  hombre.  Todas  clases  hombres  merito  ;  resumidas  cuentas,  sulpa 
suya  vizxonde  ;  pero  dire  sobrina  puede  contar  dote  viente  cinco  duros  menos,  tengo 
apartado  ;  pardiez  tamado  trabajo  atesorar-los  para  enriquecer  estrano. 

Fig.  35.  —  Ratio  of  attendance  to  enrollment  in  public  schools  of  cities  and  towns 
of  over  8,000  inhabitants  in  Ohio,  Indiana,  Illinois  and  Iowa.     N=  115. 

Fig.  36.  — Wages  of  cotton  operatives  (in  shillings  per  week),  N  \s  large,  but 
not  given.     The  data  are  taken  from  Bowley's  '  Elements  of  Statistics,'  p.  96. 

Fig.  37.  —  Age  of  graduation  from  American  colleges.  Men  only  taken. 
Ar=  1,213. 

Fig.  38.  — Cost  per  pupil  of  public  school  education  in  American  cities  of  over 
8,000  inhabitants.  The  cost  is  here  taken  per  pupil  actually  present  throughout  the 
year.  That  is,  the  cost  per  pupil  equals  amount  spent  divided  by  average  attendance. 
In  dollars.  N=  465.  The  amounts  and  average  attendances  are  those  given  in  the 
Eeport  of  the  U.  S.  Commissioner  of  Education  for  1901. 

Fig.  39. — Wages  of  American  workingmen  per  day.  In  cents.  JV=  5,123. 
The  data  are  taken  from  Bowley's  '  Elements  of  Statistics,'  p.  120.  He  quotes  them 
from  a  U.  S.  Senate  report. 

Fig.  40.  —  Figure  39  with  a  coarser  grouping. 

Fig.  41.  —  Ratio  of  attendance  to  enrollment  in  public  schools  of  American 
cities  of  over  8,000  inhabitants.     JV=545. 

Fig.  42.  —  Incomes  of  American  colleges  for  men  and  for  both  sexes.  The  five 
per  cent,  who  in  the  year  taken  had  incomes  of  over  $150,000  are  omitted.  In  thou- 
sands of  dollars.     N=  438. 

Fig.  43.  —  Age  at  marriage  of  gifted  American  men.     N=  744. 

Fig.  44.  —  Frequency  of  divorces  in  different  years  after  marriage.  The  cases 
after  twenty  years  are  undistributed  by  the  compiler  and  are  here  given  a  probable 
distribution.  ,V=  109,960.  The  data  were  taken  from  Karl  Pearson's  table,  Phil. 
Trans,  of  the  Royal  Society,  Vol.  186A,  p.  395.  He  in  turn  quotes  them  from  W.  F. 
Wilcox,  'The  Divorce  Problem.' 

Fig.  45.  — Size  of  New  England  families,  1725-1800.  The  number  of  children 
born  to  women  during  twenty  years  or  over  of  married  life.     N=  163. 

Fig.  46.  —  Infant  mortality  in  cities  and  towns  of  England  and  Wales.  Num-' 
ber  of  deaths  per  1,000  births.  N=  112.  Arranged  from  data  given  by  Miss  Clara 
Collet  in  the  Journal  of  the  Royal  Statistical  Society,  June,  1898. 

Fig.  47.  — Frequency  of  death  at  different  ages.  After  Karl  Pearson,  'Chances 
of  Death,'  Vol.  I.,  p.  27.     jVis  very  large. 

In  figures  21  to  47,  the  limits  to  which  the  surface  of  frequency  extends  are  shown 
by  short  vertical  lines  in  those  cases  where  the  length  of  the  columns  of  which  it  is 
composed  is  so  small  as  to  be  unnoticeable.     See,  for  instance,  /t  and  l2  in  Fig.  21. 

*The  13-,  14-  and  15-year  old  girls  did  not  differ  as  groups. 


IS 


MENTAL   AND  SOCIAL    MEASUREMENTS. 


24  _X  U 

15         18         21          24 

Fig.  21. — Height  of  men. 

Fig.  22. — Weight  of  men. 

Fig.  23. — Cephalic  index. 

Fig.  24. — Length  of  infants. 

Fig.  25.— Girth  of  chest. 

Fig.  26. — Strength  of  arm  pull. 

Fig.  27. — Body  temperature. 

Fig.  28. — Heart  rate  after  exercise. 


126  175 


MEASUREMENT  OF  A   GROUP. 


49 


29 


32 


30 


1 


I 


31 


34 


Fig.  29. — Keaction  time. 

Fig.  30. — Memory  span  for  digits. 

Fig.  31. — Efficiency  in  perception  of  As. 

Fig.  32. — Efficiency  in  association  of  ideas. 

Fig.  33. — Accuracy  of  estimation  of  length. 


Fig.  34. — Efficiency  in  perception  of  words. 


50 


MENTAL    ASD  SOCIAL  MEASUREMENTS. 


Jl 


\I% 


hnJL 


60  70 

35 


80  90 


36 


39 


165  a  15 


10  20 


15  165  315 

40 


38 

Fig. 

35. 

Fig. 

36. 

Fig. 

37. 

Fig. 

38. 

Fig. 

39. 

Fig. 

40. 

Fig. 

41. 

Fig 

42. 

42 

-Ratio  of  school  attendance  to  enrollment. 
-Wages  of  cotton  operatives. 
-Age  of  graduation  from  college. 
-Cost  per  pupil  of  education. 
-Wages  of  American  workingmen. 
-Wages  of  American  workingmeu. 
-Eatio  of  school  attendance  to  enrollment. 
-Incomes  of  colleges. 


MEASUREMENT  OF  A   GROUP. 


61 


1 


k 


17  27  37  47  0  4  8  12  16 

43  45 


150 


190 


47 

Fig.  43. — Age  of  marriage  of  gifted  men. 

Fig.  44. — Frequency  of  divorces  at  different  dates  after  marriage. 

Fig.  45. — Size  of  New  Englaud  families. 

Fig.  46. — Infant  mortality. 

Fig.  47. — Frequency  of  death  at  different  ages. 


52  MENTAL    AND  snd.lL   MKAsriWMENTS. 

The  quantitative  expression  of  any  group  measurement  in  a  few 
figures  capable  of  treatment  by  ordinary  statistical  methods  will  de- 
pend upon  (1)  the  considerations  already  explained  in  the  case  of 
individual  measurements,  and  also  upon  our  general  information  (2) 
about  the  group  measured  and  (3)  about  the  causes  the  action  of 
which  determines  the  quantity  measured.  A  complete  discussion  of 
(2)  and  (3)  is  impossible  because  of  the  lack  of  data,  and  even  such 
a  survey  as  the  inadequacy  of  the  data  permits  would  be  far  too  in- 
tricate and  obscure  for  the  modest  purposes  of  this  book.  All  that 
will  be  attempted  will  be  a  rough  statement  of  the  facts  about  a 
group  which  are  of  most  assistance  in  interpreting  its  surface  of 
frequency,  and  a  very  elementary  introduction  to  the  study  of  the 
relation  between  the  nature  of  the  causes  affecting  a  quantity  and  the 
quantity's  distribution.  The  former  will  be  the  subject  of  the  rest 
of  this  chapter ;  the  latter  will  be  given  in  Chapter  V. 

Hie  Interpretation  of  the  Form  of  a  Surface  of  Frequency. 

It  might  appear  reasonable  to  take  the  distribution  obtained  for 
any  group  at  its  face  value.  For  instance,  if  in  a  measure  of  the 
scholarship  of  men  one  obtained  a  distribution  like  that  represented 
in  Fig.  48,  it  might  appear  reasonable  to  say  that  intellect  was 
distributed  in  a  very  irregular  manner  and  in  such  a  way  that  there 
were  no  grades  very  far  below  the  commonest  condition,  but  that 
grades  above  it  existed  over  such  a  range  that  the  highest  ranking 
person  was  ten  times  as  far  above  the  mode  as  the  lowest  ranking 
person  was  below  it,  and  that  the  grades  up  near  the  highest  were 
more  common  than  those  a  little  nearer  the  mode.  Further  con- 
sideration, however,  might  show  that  the  infrequency  of  low  grades 
was  due  to  the  fact  that  in  our  measurements  we  had  tested  only  the 
better  classes  —  had  selected  against  the  idiots,  illiterates  and  incom- 
petents ;  and  that  the  apparently  greater  frequency  of  very  high  than 
of  moderately  high  grades  wTas  due  to  our  having  measured  some 
thousands  of  individuals  from  the  better  classes  together  with  a 
hundred  or  so  college  graduates.  Scholarship  in  general  might 
really  be  distributed  normally  as  shown  in  Fig.  49,  and  our  result 
be  due  to  the  influence  of  selection  and  of  mixing  two  species,  un- 
trained and  trained  men.  On  the  other  hand,  if  one  obtained  for 
scholarship  a  normal  distribution,  one  could  not  be  sure  that  in  the 


MEASUREMENT  OF  A    GROUP. 


53 


natural  group,  men,  scholarship  was  normally  distributed  unless 
these  same  factors  of  elimination  and  mixture  were  excluded.  For 
example,  if  one  got  a  normal  distribution  from  measuring  13-year- 
old  boys  in  the  next  to  the  last  grammar-school  grade,  he  could  be 
practically  sure  that  for  all  13-year-old  boys  the  distribution  would 
not  be  normal.  For  the  duller  13-year-old  boys  would  not  have 
reached  that  grade  and  the  very  bright  ones  would  often  have  passed 
it.  The  actual  distribution  may  be  in  part  the  result  of  the  mixture 
of  species  or  of  selection. 


FIG.  48. 


fig.  4a 
Fig.  48.  —  An  irregular  distribution  possibly  due  to  artificial  elimination  and 
mixture  of  species  in  the  course  of  the  measurements. 
Fig.  49. — A  regular  distribution. 

Homogeneous  and  Mixed   Groups. 

Homogeneity  is  in  general  not  an  absolute,  but  a  relative,  quality. 
A  group  of  animals  is  homogeneous  compared  with  a  group  of 
animals  and  plants  mixed.  A  group  of  human  beings  is  homogeneous 
compared  with  a  group  of  men,  dogs,  worms  and  fishes.  A  group 
of  college  graduates  is  homogeneous  compared  with  a  group  of  col- 
lege graduates,  illiterates  and  idiots.  Utter  homogeneity  would 
equal  identity.  We  commonly  mean  by  the  homogeneity  of  any 
group  with  respect  to  any  trait,  such  likeness  amongst  its  members, 
with  respect  to  the  forces  producing  the  trait,  that  there  is  no  reason 
for  separating  them  into  several  groups  rather  than  leaving  them  in 


:>4 


MENTAL   AND  SOCIAL  MEASUREMENTS. 


18  36  6  36 

Fig.  50.  —  Showing  six  cases  of  the  influence  of  combination  upon  the  form  of 
distribution,  viz: 

Two  normal  distributions,  A  and  B,  when  combined,  give  a  markedly  bimodal 
distribution. 

Two  normal  distributions,  C  and  D,  when  combined,  give  a  flattened  distribution. 

Four  normal  distributions,  E,  F,  G  and  H,  when  combined,  give  a  flattened  and 
positively  skewed  distribution. 

Three  normal  distributions,  2,  ./and  K,  when  combined,  give  a  markedly  skewed 
distribution. 

Two  distributions,  L  and  M,  of  identical  mode  but  differing  variability,  give, 
when  combined,  a  form  midway  between  the  two. 

Two  distributions,  JVand  O,  one  positively  and  the  other  negatively  skewed,  give, 
when  combined,  a  normal  distribution. 


MEASUREMENT  OF  A    GROUP. 


55 


17 


27 


56 


.v /:.\  y.  i  /.  a  \  / >  si  h'ia  l  m  /■:.  i  .s  i  n i:mi:xts. 


one.  Thus  the  group  '  a  Bpecies  '  of  the  zoologist  or  botanist  is  homo- 
geneous with  respect  to  its  anatomy.  Thus  the  group  'children  of 
the  Bame  race,  sex  and  age'  is  probably  homogeneous  with  respect 
to  the  trait  '  maturity. '      Thus  the  group  '  wages  of  unskilled  laborers 


10 


14 


18 


10 


14 


51 


53 


Jul 


6  10  14  18  9  14  18 

52  54 

Figs.  51,  52,  53,  54. 

under  the  same  conditions  of  work  and  cost  of  living '  is  homogeneous 
to  the  economist. 

The  effect  on  the  distribution  of  a  trait  of  putting  together  groups 
different  as  groups  with  respect  to  the  trait  can  be  seen  from  the  dia- 
grams of  Fig.  50. 

It  is  obvious,  in  general,  that  given  any  form  of  distribution,  it 
might  be  accounted  for,  so  far  as  the  bare  fact  of  its  existence  went, 
by  any  one  of  a  practically  infinite  number  of  different  compound- 


MEASUREMENT  OF  A    GROUP.  57 

iugs  of  groups.  The  mere  form  of  distribution  does  not  itself  tell. 
Recourse  must  be  had  to  a  study  of  the  real  facts  about  the  group. 

I  shall  consider  further  only  the  case  of  the  compounding  of  two 
or  more  groups,  each  of  which  by  itself  shows  approximately  normal 
distribution,  which  differ  in  respect  to  the  amount  of  the  trait.  It 
is  clear  from  the  diagrams  that  the  result  on  the  form  of  distribution 
of  the  total  group  will  be  multimodality  or  a  flattening  of  the  top  of 
the  surface  of  frequency  at  some  point.  If  one  has  reason  to  believe 
that  the  trait  he  is  studying  would  in  a  homogeneous  group  show 
normal  distribution,  the  existence  of  such  multimodality  or  flattening 
may  properly  lead  him  to  suspect  the  mixture  of  two  groups  or 
species  and  to  examine  the  cases  with  a  view  to  separating  them 
into  more  homogeneous  groups. 

One  special  case  of  importance  is  that  where  the  total  group  is  a 
compound  of  a  very  large  number  of  groups  so  differing  that  their 
central  tendencies  form  approximately  an  arithmetical  series.  Such 
total  groups  would  be,  for  instance,  measurements  of  children  eight 
to  twelve  years  of  age  in  some  physical  or  mental  trait  subject 
to  growth,  or  of  teachers'  salaries  over  a  period  of  years  during 
which  there  was  a  steady  rise  in  values.  The  death-rate  for  children 
under  a  year  reckoned  on  the  last  thirty  years'  records  in  100  cities 
would  be  a  mixture  of  thirty  different  groups. 

Selected  Groups. 
Only  very  infrequently  does  the  measurement  of  any  trait  in  a 
group  include  all  the  members  of  a  group.  It  is,  on  the  contrary, 
the  result  of  measurements  of  relatively  few  sample  individuals. 
These  represent  the  group  as  a  whole  justly  only  in  so  far  as  they 
include  the  same  percentage  of  each  grade  of  ability  in  the  group. 
Suppose  the  real  distribution  to  be  as  given  in  Fig.  51.  If  20  per 
cent,  of  each  grade  are  taken,  the  form  of  distribution,  of  course,  re- 
mains as  before.  If  20  per  cent,  of  grade  1,  18  per  cent,  of  grade 
2,  16  per  cent,  of  grade  3,  and  so  on,  are  taken,  the  form  becomes 
that  of  Fig.  52.  If  the  per  cents,  taken  are  in  order  20,  15,  10,  5, 
0,  5,  10,  15,  20,  15,  10,  5,  0,  the  form  of  distribution  becomes  that 
of  Fig.  53.  If  the  per  cents,  taken  are  in  order  0,  0,  0,  10,  20,  30, 
40,  50,  60,  70,  80,  90  and  100,  the  form  of  distribution  becomes 
that  of  Fig.  54. 


MENTAL   AND  social   MEASUREMENTS. 

In  general,  it  can  easily  be  shown  that  by  the  right  combinations 
selections  from  a  group,  a  group  with  any  form  of  distribution  can 
be  derived,  no  matter  what  the  form  of  distribution  of  the  trait  in 
the  original  group  was. 

Selection  may  occur  (1)  as  a  result  of  natural  forces  upon  a  group, 
or  (2)  as  the  result  of  unproportional  sampling  by  the  measurer. 

The  group,  living  human  beings  40  years  old,  is  thus  the  selec- 
tion by  natural  forces  from  the  group,  all  human  beings  born  40 
years  ago,  a  selection,  to  some  extent  at  least,  of  the  physically  more 
vigorous,  morally  less  murderous,  and  so  on. 

The  group,  seventeen-year-old  boys  measured  in  school,  is  a  selec- 
tion from  all  boys  seventeen  years  old,  due  to  the  measurer's  will- 
ingness to  take  boys  not  absolutely  at  random,  but  as  found 
conveniently.  The  selection  is,  to  some  extent  at  least,  of  the  more 
ambitious  and  gifted  intellectually. 

The  influence  of  nature  in  changing  the  distribution  of  a  trait  in 
a  group  by  selecting  for  survival  on  the  basis  of  the  trait's  amount 
is  one  of  the  most  important  topics  for  science,  but  does  not  need 
further  mention  here.  The  influence  of  circumstances  in  providing 
the  student  with  a  set  of  selected  samples  the  distribution  of  which 
is  unlike  that  of  the  total  group  the  student  takes  them  to  represent, 
is,  on  the  other  hand,  the  most  important  cause  of  the  majority  of 
statistical  fallacies  in  the  mental  sciences,  and  requires  discussion 
here  and  in  another  connection  later. 

Although  any  form  of  frequency  surface  may  be  derived  from  any 
other  by  the  proper  method  of  selection  of  cases,  and  although,  con- 
sequently, from  the  actual  form  of  a  surface  of  frequency  nothing  can 
be  concluded  concerning  the  group  from  which  it  represents  a  selec- 
tion unless  the  method  of  selection  is  known,  yet  certain  appearances 
may  well  serve  to  awaken  suspicion  and  guide  the  student  to  inter- 
pret the  measurements.  In  particular,  skewness  is  so  often  con- 
nected with  picking  for  study  extreme  cases  of  a  group  which  as  a 
whole  would  give  approximately  normal  distribution,  that  it  is  cer- 
tainly advisable  always,  when  confronted  by  a  group  measure  show- 
ing skew  distribution,  to  ascertain  whether  the  group  is  not  a  par- 
tial picking  from  a  normally  distributed  total  group. 

The  reader  who  has  carefully  attended  to  the  numerous  theoretical 
reservations  and  cautions  of  this  chapter  will  now7  be  able  to  use, 


MEASUREMENT  OF  A    GROUP.  59 

and  not  abuse,  the  general  practical  advice  to  which  it  leads,  which  is : 
If  for  any  reason  you  have  to  make  an  hypothesis  about  the  form 
of  distribution  of  any  trait  in  the  absence  of  the  facts,  the  most 
likely  one  in  the  case  of  anatomical,  physiological  and  mental  traits 
is  that  the  form  will  be  something  like  the  normal  probability  sur- 
face. The  probable  error  of  any  such  hypothesis  is  least  in  the  case 
of  anatomical  traits.  Prediction  of  the  form  of  distribution  of 
economic  traits  is  very  insecure.  The  interpretation  of  any  ascer- 
tained form  of  distribution  is  difficult,  but  may  prove  very  instructive. 
If  in  dealing  with  group  measurements  you  can,  without  violating 
any  knoivn  fact,  use  the  hypothesis  that  in  a  homogeneous  group  not 
subject  to  selection  on  the  basis  of  the  trait  in  question,  any  mental 
trait  due  to  natural  as  opposed  to  artificial  causes,  is  distributed  ap- 
proximately normally,  do  so. 

The  normal  surface  of  frequency  (which  is  that  of  a  quantity  due 
to  the  chance  combinations  of  n  causes,  all  equal  and  independent, 
when  n  is  infinitely  large)  is,  as  stated  on  page  36,  the  surface  enclosed 
by  the  normal  probability  curve, 

/  =*. 

(    Y=  Pe  2nP'i      or     y  =  e   *", 

or  some  specialized  form,  as 


1        yr\ 

=  — -=e^2  ) 

u\/2n  / 


V     pS2 

and  the  abscissa  or  base  line  on  which  x  is  scaled. 

In  this  form  of  distribution  the  Average,  Median  and  Mode  coin- 
cide, for  y  is  the  same  for  any  given  —  x  as  for  the  same  -f  x,  and 
is  greatest  when  x  =  0.  Constant  relations  hold  between  the  differ- 
ent measures  of  variability,  viz  : 

<r=  1.25331  A.  D. 
rr=  1.4825  P.  E. 
A.  D.  =    .7979  a 
A.  D.  =  1.1843  P.  E. 
P.  E.  =    .6745  a 
P.  E.  =    .8453  A.  D. 
Between  Av.  —  a  and  Av.  +  a  are  68.2  per  cent,  of  the  cases. 

"       Av.  =  A.  D.  and  Av.  +  A.  D.  are  59.5  per  cent,  of  the  cases. 
"       Av.  —  P.  E.  "    Av.  +  P.  E.  "    50       "     "      "    "      " 


\  /'  I  I     AND   S0(  7.1  /.    MKA.s  I  TJ-JMEXTS. 


The  frequencies  of  different  deviations  from  the  mode  (or  average 
or  median)  are  given  in  gross  in  Tables  XXII.  and  XXIII.  De- 
tailed tables  will  be  given  later. 


TABLE  XXII. 

I  notes  of  Deviations  Above  the  Mode  in  a  Normal  Surface  of 
Frequency  in  Terms  oe  A.  D.    The  Figures  can  be  Used 
Identically  for  Minus  Deviations. 

Between  +     0  and  +    .2  A.  D.  are  6.3    per  cent,  of  the  cases. 


-       -    -2 

a 

+    -4 

'    6.2 

«       +    .4 

(i 

+    -6 

'    5.9 

"       +   .6 

It 

+    -8 

1   5.4 

"       +    .8 

it 

+  1.0 

<   5.0 

"       +1.0 

a 

f  1.2 

'   4.3 

"       +1.2 

tt 

+  1.4 

'    3.7 

"       +1.4 

it 

+  1.6 

'    3.1 

"       +1.6 

it 

+  1.8 

'    2.6 

"       +1.8 

it 

+  2.0 

'   2.0 

"       +2.0 

a 

+  2.2 

'    1.5 

"       +2.2 

it 

+  2.4 

'    1.2 

"       +2A 

a 

+  2.6 

'    0.9 

"       +2.6 

it 

+  2.8 

'    0.6 

"       +2.8 

a 

+  3.0 

'    0.5 

"       +3.0 

i( 

+  3.2 

'    0.26 

"       +3.2 

it 

+  3.4 

'   0.21 

"       +3.4 

it 

+  3.6 

1    0.13 

"       +3.6 

a 

+  3.8 

'   0.07 

"       +3.8 

a 

+  4.0 

'   0.06 

"       +4.0 

it 

+  4.2 

'    0.03 

"       +4.2 

a 

+  4.4 

'   0.02 

"       +4.4 

a 

+  4.6 

'    0.01 

"       +4.6 

it 

+  <*> 

'    0.01 

TABLE  XXIII. 
Frequencies  of  Plus  Deviations  or  of  Minus  Deviations.     In  Terms  of  a. 


Between 


0  and 

•  2cr  ar« 

7.93  per 

.2 

a 

.4 

it 

7.61 

.4 

it 

.6 

a 

7.04 

.6 

" 

.8 

tt 

6.24 

.8 

it 

1.0 

1 1 

5.32 

1.0 

it 

1.2 

tt 

4.36 

1.2 

it 

1.4 

tt 

3.43 

1.4 

it 

1.6 

tt 

2.59 

1.6 

it 

1.8 

tt 

1.89 

1.8 

a 

2.0 

tt 

1.32 

2.0 

it 

2.2 

tt 

0.88 

2.2 

it 

2.4 

it 

0.57 

2.4 

it 

2.6 

tt 

0.35 

2.6 

it 

2.8 

tt 

0.22 

2.8 

n 

3.0 

a 

0.12 

3.0 

it 

3.2 

it 

0.06 

3.2 

a 

3.4 

n 

0.04 

3.4 

it 

3.6 

a 

0.02 

3.6 

tt 

■x, 

a 

0.01 

cent,  of  the  cases. 


CHAPTER   V. 

THE     CAUSES     OF     VARIABILITY     AND     THE    APPLICATION    OF     THE 
THEORY    OF    PROBABILITY    TO    MENTAL    MEASUREMENTS. 

The  varying  measures  of  one  individual's  performances  and  the 
varying  measures  of  the  individuals  in  a  group  were  found  in  Chap- 
ters III.  and  IV.  to  be  often  distributed  approximately  after  the 
fashion  of  the  surface  of  frequency  enclosed  by  the  probability  curve 
and  its  abscissa.  In  those  chapters  brief  mention  was  made  of  the 
properties  of  this  surface  or  type  of  distribution,  acquaintance  with 
which  is  a  great  assistance  to  convenient  handling  of  mental  measure- 
ments. The  recognition  of  this  type  of  frequency  surface,  the  appre- 
ciation of  its  meaning  and  that  of  certain  common  departures  from  it, 
and  the  use  of  tables  derived  from  the  probability  integral  in  calcu- 
lations of  measurements  of  traits  approximately  normally  distributed, 
are  all  possible,  at  least  to  the  moderate  degree  required  for  ordinary 
statistical  work,  without  any  knowledge  of  the  abstract  principles 
involved.  But  such  knowledge  is  well  worth  obtaining  for  the  sake 
of  the  additional  insight  into  the  meaning  of  concrete  facts  thereby 
given,  and  even  merely  for  the  sake  of  the  additional  facility  in  the 
use  and  construction  of  tables  and  the  common  formula?.  The  pres- 
ent chapter  will,  therefore,  contain  a  very  simple  introduction  to  the 
study  of  the  applications  of  the  mathematics  of  probability  to  the 
theory  of  the  distribution  of  mental  traits.  From  it  the  student  may 
proceed  to  the  study  itself  with  the  aid  of  the  references  given  at  the 
end  of  the  chapter.  The  chapter  will  also  introduce  the  student  to 
the  more  general  problem  of  the  relation  which  the  nature  of  the 
causes  determining  the  amount  of  a  trait  hold  to  the  trait's  dis- 
tribution. 

Let  us  begin  with  the  consideration  of  a  quantity  which  is  de- 
pendent on  the  action  of  one  cause  which  is  as  likely  to  occur  as  not, 
and  call  the  cause  a.  For  example,  a  may  be  the  action  of  John's 
father  in  giving  him  a  Christmas  gift  of  a  dollar. 

The  condition  of  affairs  resulting  will  be,  of  course,  no  action  or 
a.  The  quantity  in  question,  John's  Christmas  money,  will  be  0  or 
$1.00.     Its  distribution  will  be 

61 


62  MENTAL  AND  SOCIAL  MEASUREMENTS. 

Quantity.  Frequency. 

Dollars.  Per  cent. 

0  50 

1  50 

It-  surface  of  frequency  will  be  a  rectangle,  composed  of  two 
rectangles  of  equal  base  and  altitude. 

Suppose  now  that  two  causes  contribute  to  determine  the  quantity, 
a  and  A,  the  possible  actions  of  John's  father  and  mother,  and  that 
all  combinations  of  these  causes  are  equally  likely.  The  condition 
of  affairs  resulting  will  be,  then,  no  action,  a,  b  or  ab,  all  being 
equally  likely.  If  now  a  =  a  gift  of  §1.00  and  6  likewise,  the  quan- 
tity in  question,  John's  Christmas  money,  will  be  0,  $1.00,  $1.00 
or  $2.00.     Its  distribution  will  be 

Quantity.  Frequency. 

Dollars.  Per  cent. 

0  25 

1  50 

2  25 

Its  surface  of  frequency  is  that  shown  in  Fig.  55.  If  the  condi- 
tions are  kept  the  same  but  the  number  of  causes  increased  to  three, 
the  condition  of  affairs  will  be,  no  action,  a,  b,  c,  ab,  ac,  be,  or  abc. 
If  as  before  a  =  b  =  c  in  magnitude,  John  will  get  $2.00  as  often  as 
$1.00  and  three  times  as  often  as  nothing  or  $3.00. 

The  surface  of  frequency  of  the  quantity,  John's  Christmas  money, 
will  be  four  rectangles,  as  shown  in  Fig.  5Q. 

Keeping  all  the  conditions  the  same,  let  the  number  of  causes  be 
increased  to  4,  then  to  5,  and  then  to  6.  The  condition  of  affairs  in 
each  case  and  the  resulting  distribution-schemes  and  surfaces  of  fre- 
quency are  given  in  Tables  XXIV.,  XXV.  and  XXVL,  and  Figs. 
57,  58  and  59. 

TABLE  XXIV. 
Combinations  of  4  Causes,  a,  b,  c  and  d. 


bd,     cd 


a,         b, 

c, 

d 

ab,       ac, 

ad, 

be, 

abc,     abd, 

acd, 

bed 

abed 

Value  in 
Dollars. 

Probable 
Frequency, 

0 

1 

1.00 

4 

2.00 

6 

3.00 

4 

4.00 

1 

CAUSES  OF  VARIABILITY.  63 

TABLE  XXV. 

Combinations  of  5  Causes,  a,  b,  c,  d  and  e. 

Value  in        Probable 
Dollars.      Frequency. 

0  0  1 

a,  b,  c,  d,  e  1.00  5 

ab,  ac,  ad,  ae,  be,  b<l,       be,       cd,      ce,       de            2.00  10 

abc,  abd,  abe,  acd,  ace,  ade,     bed,     bee,     bde,     cde          3.00  10 

abed  abce,  abde,  acde,  bede  4.00  5 

abede  5.00  1 

TABLE  XXVI. 

Combinations  of  6  causes,  a,  b,  c,  d,  e  and/. 

Value  in  Probable 

Dollars.         Frequency- 

0  0  1 

a,  b,  c,  d,  e,         f  1.00  6 

ab,  ac,  ad,  ae,  af,  be,        bd,         be 

bf,  cd,  ce,  cf,  de,         df,        ef  2.00  15 

abc,  abd,  abe,  abf,  acd,      ace,       acf 

ade,  adf,  aef,  bed,  bee,        bef,        bde 

bdf,  bef,  cde,  cdf,  cef,        def  3.00  20 

abed,  abce,  dbef,  abde,  abdf 

abef,  acde,  aedf,  acef,  adef 

bede,  bed/,  beef,  bdef,  cdef  4.00  15 

abode,  abedf,  abcef,  abdef 

acdef  bedef  5.00  6 

abedef  6.00  1 

TABLE  XXVII. 

Combinations  of  10,  15  and  20  Causes. 

Quantity.  Frequency  in  case. 

Dollars.                         Of  10.                             Of  15.  Of  20. 

Oil  1 

1  10                                  15  20 

2  45                             105  200 

3  120  455  1,080 

4  210  1,365  4,505 

5  252  3,003  14,944 

6  210  5,005  38,370 

7  120  6,435  77,420 

8  45  6,435  125,970 

9  10  5,005  167,960 

10  1  3,003  184,756 

11  1,365  167,960 

12  455  125,970 

13  105  77,420 

14  15  38,370 

15  1  14,944 

16  4,505 

17  1,080 

18  200 

19  20 
%        20  1 

21 


04  MENTAL   AND  social   MEASUREMENTS. 

It  is  apparent  that  the  surface  of  frequency  of  a  quantity  depend- 
ent upon  the  action  of  causes  equal  in  magnitude,  any  combination 
of  which  is  equally  probable,  tends,  as  the  number  of  these  causes 
becomes  great,  to  approach  the  type  we  often  find  in  the  case  of  ana- 
tomical traits.  This  is  emphasized  by  Table  XXVII.  and  Figs.  60, 
61  and  62,  which  give  the  results  in  our  illustration  if  the  number 
ot^  causes  is  increased  to  10,  15  and  20  respectively.  When  the 
number  of  causes  is  very,  very  great  the  result  is  the  normal  proba- 
bility surface  (Fig.  63). 

The  normal  type  of  distribution  may  therefore  be  expected  in  the  ' 
case  of  the  different  performances  or  measures  of  an  individual  in  the 
>arue  trait,  if  any  one  of  his  performances  in  the  trait  is  due  to  the 
action  of  some  one  combination  from  a  large  number  of  causes  of 
equal  magnitude  which  are  independent  of  each  other,  so  that  any 
combination  is  as  likely  to  occur  as  any  other ;  may  be  expected  in 
the  case  of  the  different  measures  of  individuals  in  a  group,  if  the 
tendency  of  any  individual  in  the  trait  is  due  to  the  action  of  some 
one  combination,  characteristic  of  his  make-up,  from  such  a  large 
number  of  causes.  If,  that  is,  we  think  of  any  single  act  of  a  per- 
sou  as  a  result  of  a  chance  combination  from  amongst  a  number  of 
causes  which  determine  acts  of  that  sort  characteristic  of  him,  we 
shall  expect  his  different  manifestations  of  the  trait  of  which  that  act 
is  a  sample  to  be  normally  distributed ;  so  also,  if  we  think  of  the 
quantity  of  a  trait  in  any  single  individual  of  a  group  as  a  result  of 
a  chance  combination  from  amongst  a  number  of  causes  characteristic 
of  the  group  as  a  whole  which  determine  that  trait,  we  shall  expect 
the  manifestations  of  that  trait  by  the  group  of  which  he  is  a  sample 
to  be  normally  distributed. 

The  clause  'so  that  any  combination  is  as  likely  to  occur  as 
another '  and  its  synonymous  phrase  '  a  chance  combination  from 
amongst'  need  some  explanation.  They  refer  to  the  fact  that  the 
causes  must  be  independent  of  each  other  if  the  distribution  of  the 
trait  is  to  be  normal.  The  need  of  this  condition  will  be  apparent 
from  a  concrete  illustration. 

Suppose  that  in  our  previous  case  of  John's  Christmas  money  the 
six  causes  a,  b,  c,  d,  e  and  /  were  as  .before,  except  that  no  action 
was  barred  out,  and  that  if  a  acted  6  and  c  must  also,  and  d,  e  and  / 
could  not ;  while  if  d  acted  e  and  /  must,  but  a,  b  and  c  could  not. 


CAUSES   OF    VARIABILITY. 


65 


0         12 

55 


o 
56 


o 
58 


3  0 


59 


J~L 


o 
57 


60 

Figs.  55,  56,  57,  58,  59,  60. 


66  MENTAL    AND   SOCIAL    MEASTUEMENTS. 


61 


n 


64 

6? 

Figs.  61,  62,  63,  64,  65,  66,  67. 


CAUSES   OF   VARIABILITY.  67 

Imagine,  for  instance,  that  it  was  agreed  to  take  turns  in  preventing 
a  penniless  Christmas ;  that  the  father  agreed  to  give  his  dollar  if  the 
mother  and  sister  would  always  join  with  him  and  the  grandfather, 
grandmother  and  brother  would  keep  their  money  to  themselves, 
while  the  grandfather  agreed  to  give  his  dollar  upon  the  condition 
that  he  be  joined  by  grandmother  and  brother  and  that  father,  mother 
and  sister  refrain.  The  condition  of  affairs  then  could  only  be  abc 
or  def  instead  of  the  range  of  possibility  of  the  illustration  in  its  first 
form.  Although  there  are  six  causes,  the  result  is  as  if  there  were 
only  one,  and  that  always  operative. 

Suppose  the  presence  of  a  or  b  or  c  to  always  cause  that  of  the 
other  two  of  the  three,  and  similarly  for  the  presence  of  d,  e  or  /. 
This  means  that  whenever  cause  a  appears  it  adds  to  itself  6  and  c, 
whenever  b  appears  it  adds  to  itself  a  and  c,  and  so  on.  Every  con- 
dition in  Table  XXVI.  with  a  or  b  or  c  in  it  must  then  become  abc; 
every  condition  with  d  or  e  or  /  must  become  def  ;  every  condition 
with  one  from  the  abc  and  one  from  the  def  group  must  become  abcdef. 
Thus  the  condition  of  affairs  would  be,  instead  of  that  in  Table 
XXV L,  the  following  :  no  action,  1  ;  abc,  7  ;  def,  7 ;  abcdef,  49. 

The  distribution  would  then  be  (as  shown  in  Fig.  64)  : 

Quantity.  Frequency. 

Dollars. 

0  1 

3  14 

6  49 

Suppose  the  presence  of  a  to  imply  always  that  of  e,  d,  e  and  /, 
the  presence  of  b  to  imply  always  that  of  d,  e  and  /,  the  presence  of 
c  to  imply  that  of  e  and/,  and  the  presence  of  (/  that  of/.  The  dis- 
tribution would  be  (as  shown  in  Fig.  65) : 

Quantity.  Frequency. 

Dollars. 

0  1 

1  2 

2  3 

3  6 

4  12 

5  24 

6  16 

Suppose  the  presence  of  a  or  b  or  c  implies  the  other  two  of  the 
three,  and  that  the  presence  of  e  implies  that  of  /,  and  vice  versa. 
The  distribution  will  be  (as  shown  in  Fig.  66) : 


68  MENTAL   AND  social    MEASUREMENTS. 

Quantity.  Frequency. 

Dollars. 

0  1 

1  1 

2  3 

3  10 

•1  7 

5  19 

6  23 

It  is  clear  then  that  the  interdependence  of  the  causes  determin- 
ing the  quantity  of  a  trait  may  cause  all  sorts  of  departures  from  the 
normal  type  of  distribution,  skewnesses  and  multimodal  conditions, 
etc. ;  may,  in  less  technical  terms,  cause  the  amounts  of  it  appearing 
in  an  individual's  different  records  or  in  the  different  individuals  of 
a  group  to  vary  in  all  sorts  of  ways.  In  the  illustration  only  simple 
and  total  dependencies  were  considered.  Complex  and  partial  de- 
pendencies would  complicate  the  results  to  a  well-nigh  endless  extent. 

It  should,  however,  be  noted  that  if  the  causes  are  numerous  and 
their  interdependences  of  a  random,  hit-or-miss  character,  their  com- 
bined action  may  be  practically  identical  with  that  of  totally  independ- 
ent causes.  Thus,  to  continue  with  the  same  illustration,  if  there 
were  five  hundred  relatives  they  might  plan  together  in  groups  on 
various  ways  to  give  or  withhold,  and  yet  the  final  resultant,  the 
probable  total  of  John's  Christmas  income,  might  show  no  consider- 
able differences  from  the  total  in  case  they  had  all  acted  independently. 

A  similar  principle  holds  with  reference  to  the  equivalence  of  the 
causes  in  amount.  In  our  illustration  we  demanded  perfect  equality, 
and  a  little  experimentation  will  convince  the  reader  that  the  ap- 
proximation to  the  normal  surface  of  frequency  tends  to  disappear  if 
(i  >  b  >  c  >  (1,  etc.*     However,  with  many  causes  and  with  a  not  too 

*For  instance,  let  the  cause  a  equal  10,  b  equal  5  and  c,  d,  e  and /each  equal  1. 
Then  instead  of  the  distribution  of  Table  XXVI.  we  have  (as  shown  in  Fig.  67): 


Quantity. 
Dollars. 

Frequency. 

0 

1 

1 

4 

2 

6 

3 

4 

4 

1 

5 

1 

6 

4 

7 

6 

8 

4 

9 

1 

Quantity. 
Dollars. 

Frequency. 

10 

1 

11 

4 

12 

6 

13 

4 

14 

1 

15 

1 

16 

4 

17 

6 

18 

4 

19 

1 

CAUSES  OF   VARIABILITY.  69 

great  variation  in  the  amounts,  the  resulting  distribution  may  closely 
mimic  the  perfectly  normal  type. 

Finally,  it  should  be  remembered  that  the  illustration  taken  is 
untrue  to  the  common  conditions  of  life  in  one  respect.  For  these 
show  us,  not  a  group  of  causes,  a  chance  combination  from  which  de- 
termines the  event,  but  such  a  group  acting  together  with  some  constant 
cause  or  set  of  causes.  Stature,  strength,  memory,  wage-earning 
capacity,  are  due  to  certain  constant  causes  which  always  act  on  all, 
plus  a  group,  the  action  of  which  may  be  regarded  in  the  mathemat- 
ical fashion  of  this  chapter.  The  addition  of  such  a  constant  set  of 
causes  does  not,  of  course,  alter  the  form  of  distribution  in  the  least, 
but  simply  adds  the  same  amount  to  all  its  quantities,  pushes  them 
all  ahead  on  the  scale.  In  our  illustration  the  «'s,  6's,  c's,  etc., 
might  more  properly  be  the  amounts  which  different  friends  might 
or  might  not  give  in  addition  to  minimum  sums,  k,  kv  k2,  etc.,  which 
they  always  give,  or  be  the  gifts  of  some  friends,  who  could  not  be 
counted  on,  superadded  to  a  set  of  inevitable  gifts  x,  y,  z,  etc.,  from 
a  few. 

The  commoner  method  of  describing  the  type  of  causation  result- 
ing in  a  normal  surface  of  frequency  of  the  amount  of  a  trait  starts 
with  the  presupposition  that  a  certain  amount  tends  to  be  and  con- 
siders the  causes  as  increasing  or  decreasing  this.  It  is  also  common 
to  use  the  frequencies,  not  of  amounts  of  some  continuous  quantity, 
but  of  different  proportions  of  black  to  white,  or  the  like,  in  a  chance 
draw  of  balls.  The  principles  involved  are  precisely  the  same  as 
those  which  have  appeared  in  the  more  readily  understood  cases  used 
here. 

I  have  so  far  tried  especially  to  show  how  the  cooperation  of  a 
number  of  causes,  each  of  which  has  a  given  likelihood  of  acting, 
may  produce  in  the  trait  due  to  them  a  distribution  of  the  so-called 
normal  type.  Incidentally,  it  has  been  noted  that  in  general  the 
form  of  distribution  of  any  variable  trait  is  due  to  the  number  of 
causes  that  influence  its  amount,  their  magnitude  and  their  interre- 
lations. 

The  form  of  distribution  then  is  purely  a  secondary  result  of  a 
trait's  causation.  There  is  no  typical  form  or  true  form.  There  is 
nothing  arbitrary  or  mysterious  about  variability  which  makes  the 
normal  type  of  distribution  a  necessity,  or  any  more  rational  than 


70  MENTAL  AM)  social  m  i:\sri;  i:m  i:  xrs. 

any  other  sort  or  oven  any  more  to  be  expected  on  a  priori  grounds. 
Nature  does  not  abhor  irregular  distributions. 

On  "  priori  grounds,  indeed,  the  probability  curve  distribution 
would  be  exactly  shown  in  any  actual  trait  only  by  chance.  For 
only  by  chance  would  the  necessary  conditions  as  to  causation  be  ful- 
filled. And  in  point  of  fact,  as  the  reader  has  constantly  been  told 
liy  the  adjective  'approximate/  the  exact  probability  curve  distribu- 
tion does  not  appear  in  the  facts  or  give  signs  of  being  at  the  bottom 
of  the  facts  of  mental  life.  The  common  occurrence  of  distributions 
approaching  it  is  due,  not  to  any  wonderful  tendency  of  a  group  of 
cooperating  causes  to  act  so  as  to  mimic  the  combinations  of  mathe- 
matical quantities  equal  and  equally  probable,  but  to  the  fact  that 
many  traits  in  human  life  are  due  to  certain  constant  causes  plus 
many  occasional  causes  largely  unrelated,  small  in  amount  in  com- 
parison with,  the  constant  causes  and  of  the  same  order  of  magnitude 
among  themselves. 

It  is  the  folly  of  the  ignoramus  in  statistics  to  neglect  the  applica- 
tion of  the  algebraic  laws  of  combinations  to  variable  phenomena  ;  it 
would  be  the  folly  of  the  pedant  to  try  to  bend  all  the  variety  of 
nature  into  conformity  with  one  particular  case  of  the  frequency  of 
combinations.* 

The  student  interested  in  this  subject  should  read  some  standard 

account  of  the  algebra  of  combinations  and  probability,  and  Part  II. 

of  Bowley's  '  Elements  of  Statistics.'     Further  references  will  be 

found  on  page  327  of  the  latter. 

*  It  is  a  question  whether  students  of  mental  measurement  should  not  from  the 
beginning  be  taught  to  put  the  so-called  normal  distribution  in  its  proper  place  as 
simply  one  amongst  an  endless  number  of  possible  distributions,  each  and  all  due  to 
and  explainable  by  the  nature  of  the  causes  determining  the  variations  in  the  trait. 
The  frequency  of  the  occurrence  of  distributions  somewhat  like  it  could  then  be  ex- 
plained by  a  vera  causa,  the  frequency  of  certain  sorts  of  causation.  On  general  prin- 
ciples this  seems  desirable,  but  in  order  to  make  for  the  student  connections  with  the 
common  discussions  of  statistical  theory  and  practice  and  with  the  concrete  work 
that  has  been  done  with  mental  measurements,  I  have  compromised  and  subordi- 
nated the  general  rationale  of  the  form  of  distribution  to  the  explanation  of  the 
probability  curve  type. 


CHAPTER   VI. 

THE    AEITHMETIC    OF    CALCULATING    CENTRAL    TENDENCIES 
AND    VARIABILITIES. 

The  arithmetic  of  calculating  averages,  medians,  modes,  a's, 
A.  D.'s,  P.  E.'s  and  other  measures  of  central  tendency  and  of  varia- 
bility is  simple  and  straightforward  if  one  bears  in  mind  (1)  that 
mental  and  social  quantities  are  commonly  continuous,  so  that  any 
figure  given  as  a  measure  means  not  a  point,  but  a  distance  on  the 
scale,  and  (2)  that  this  distance  is  often  that  from  the  given  figure  to 
the  next  figure,  so  that  the  real  value  of  the  figure  is  itself  plus  one 
half  of  the  unit  of  the  scale. 

The  short  methods  of  obtaining  averages,  a's  and  A.  D.'s  by  guess- 
ing at  the  value  and  then  correcting,  are,  however,  foreign  to  the 
mathematical  habits  of  one's  school  days  and  ordinarily  require  sys- 
tematic practice  before  one  gains  surety  and  facility  in  their  use.  It 
will  probably  be  advisable  for  the  student  to  test  himself  with  many 
simple  problems,  proving  his  result  by  the  use  of  the  longer  tradi- 
tional methods.  In  this  and  later  mumerical  work  it  will  be  of  as- 
sistance to  have  at  hand  Crelle's  '  Rechentafeln,''  which  enable  one  to 
multiply  and  divide  by  numbers  up  to  1,000  with  no  labor  save  for 
eyes  and  fingers,  and  Barlow's  l  Tables/  which  give  the  squares  and 
square  roots  of  all  numbers  up  to  10,000. 

The  labor  of  calculating  averages  can  be  much  reduced  by  adopt- 
ing the  method  which  most  of  us  would  probably  use  in  a  case  like 
this  :  To  get  the  average  of  54,  52,  64,  56  and  50.  Remembering 
that  the  average  is  such  a  figure  that  the  sum  of  differences  between 
it  and  the  measures  above  it  is  equal  to  the  sum  of  the  differences 
between  it  and  the  measures  below  it,  one  takes  56  as  the  average. 
The  differences  below  are  2,  4  and  6,  that  above  is  8.  If  the  average 
was  altered  by  —  .8,  or  to  55.2,  the  differences  below  would  be  1.2, 
3.2  and  5.2,  and  those  above  would  be  8.8  and  .8.  This  common 
procedure  consists  in  guessing  at  an  approximate  average  and  then 
correcting  it  from  knowledge  of  the  sums  of  the  minus  and  plus  de- 
viations from  it.     It  lets  us  add  small  numbers  instead  of  large  and, 

71 


72 


.1/ EX  TAL    A  ND  S0(  7. 1  L   ME.  1 S I TREMENTS. 


a-  will  he  Been,  Liivcs  us  at  the  same  time  as  the  average,  an  approxi- 
mate measure  of  the  average  deviation  from  it. 

Tlu'  choice  of  an  approximate  average  is  commonly  easy  after  an 
inspection  of  the  total  distribution,  and  one  soon  acquires  skill  in 
making  a  correct  choice  in  any  case. 

Suppose  the  measures  to  be  as  follows: 

Reaction-Tim i:s  or  V.  H. 


Quantity. 

Frequency 

Seconds. 

120-124.99  or  .1225 

2 

125 

3 

130 

11 

135 

13 

140 

11 

145 

13 

150 

7 

155 

8 

160 

13 

165 

8 

170 

1 

175 

3 

180 

3 

185 

0 

190 

0 

195 

1 

Either  .145  -  .1499  (i.  e.,  .1475)  or  .150  -  .154  (L  e.,  .1525) 
would  do  for  a  guess.  I  will  use  .145  —  .1499.  We  have  then  to 
obtain  the  minus  and  plus  deviations  from  .1475,  the  central  point 
of  the  .145  —  .1499  group.  To  save  labor  in  multiplication  and 
addition  I  shall  measure  these  in  terms,  not  of  units  of  the  scale,  but 
of  steps  of  the  scale,  i.  e.,  using  five  thousandths  of  a  second  as  the 
unit.     We  have  then  for  minus  and  plus  deviations  : 


2  deviations  of  —  5  or  —  10 


deviations  of 


3 
11 
13 
11 
40 


—  4  or  — 12 

—  3  or  —  33 

—  2  or  — 26 

—  1  or  — 11 

-92 


8 
13 
8 
1 
3 
3 
0 
0 

J^ 
44 


+  1  or  +  7 
+  2  or  +  16 
+  3  or  +  39 
+  4  or  +  32 
+  5  or  +  5 
+  6  or  +  18 
+  7  or  -(-  21 


+  10  or  +  10 

+  148 


METHODS  OF  CALCULATION. 


73 


The  approximate  average  is  evidently  too  low.  It  can  be  cor- 
rected by  adding  to  it  the  algebraic  sum  of  the  deviations  divided 
by  the  number  of  cases.  In  the  illustration  this  will  be  -f  ||-  or 
+  .58.  .58  of  a  step  =  2.9  thousanths  of  a  second.  The  corrected 
average  is  then  .1475  -f-  .0029  or  .1504  sec.  Calling  the  algebraic 
sum  of  the  deviations  divided  by  the  number  of  cases  d.lcL  av._approx.  av., 
we  may  summarize  this  whole  calculation  in  the  formulae  : 


Av. 


Av 


+  da 


d 


act,  av.— approx.  av. 


approx.     i     "'act,  av.  —  approx .  av. 

2a;  (algebraic) 
n 


Determination  of  the  Mode. 

In  determining  the  mode  one  should  seek  not  only  the  measure  that 
is  the  most  frequent  on  the  basis  of  the  limited  series  of  measures  he 
has  before  him,  but  also  the  one  that  would  probably  be  the  most 
frequent  if  a  very  great  number  of  measures  were  at  hand.  There 
are  two  convenient  tests  of  the  latter  fact.  The  mode  from  an  infinite 
series  of  measures  will  probably  be  a  measure  representing  the  acme 
or  culmination  of  a  somewhat  steady  tendency  of  neighboring 
measures  to  greater  and  greater  frequency.  Graphically  speaking 
it  will  be  the  apex  of  a  slope.  Hence  we  may  consider  the  general 
tendency  of  the  surface  as  a  whole  to  rise  to  a  maximum,  grouping 
the  cases  so  as  to  show  a  fairly  regular  rise,  and  use  this  knowledge 
in  deciding  the  probable  mode. 

Doing  this  in  the  present  case  we  get : 


Ability. 
115-124.99 

Frequency. 
2 

Ability. 
120-134.99 

Frequency. 
16 

125 

14 

135 

37 

135 

24 

150 

28 

145 

20 

165 

12 

155 

23 

180 

3 

1G5 

9 

195 

1 

175 

6 

185 

0 

195 

1 

.145  up  to  .150  is  probably  the  best  choice  for  a  mode. 

The  mode  may  be  obtained  from  a  quarter,  then  from  a  half, 
then  from  three  quarters  of  the  cases  taken  at  random,  and  the  in- 
fluence of  the  increase  in  number  of  cases  upon  the  position  taken  by 


7t  MENTAL  Axn  social  m  i:.wr  i:  i:m  i:xts. 

tlu>  mode  may  be  used  to  prophesy  what  position  it  would  probably 
take  with  a  very  groat  number  of  cases. 

( lommonly  with  200  or  more  measurements  and  with  a  grouping 
into  not  over  18  divisions,  the  mode  is  clear  enough.* 

The  series  of  measures  of  Table  XXVIII.  may  be  taken  as  an 
example.  26  to  30  is  the  choice  for  a  broad  mode  and  28  to  30 
the  best  choice  for  a  narrow  one. 

TABLE  XXVIII. 

Money  Available  for  School  Purposes  Divided  by  Average  Attendance  ; 

that  is,  Cost  per  Pupil  for  Full  Year's  Actual  Attendance. 

Cities  of  U.  S.     Keport  of  Com.  of  Ed.,  1901. 

Quantity .        Frenuencv  Frequency  in         Quantity.        Freauencv  Frequency  in 

Dollars.         frequency.        wider  grouping.         Dollars.         rrequencj.        wider  grouping. 

10-11.99  6  n                       4  5                         9 

12-  5  6  4 

14  10  n                      8  6                         9 

6  14  "                       60  3 

8  20  36                       2  2                         6 

20  16  4  4 

2  31  6Q                        6  5                          9 

4  29  8  4 

6  34  73                     70  2                         2 

8  39  2  0 

30  31  61                        4.2  2 

2  30  6  0 

4  24  42                       8  2                         3 

6  18  80  1 

8  17  39                        2  2                          2 

40  22  4  0 

2  16  39                        6  0                          Q 

4  16  8  0 

6  15  29  90  X  2 

8  14  2  1 

50  3  ,      ,Q  4  0  , 

2  10  13  6  J_ 

465 

Determination  of  the  Median.  —  The  median  is  the  [  (n  -j-  1)  2]"' 
measure. 

Count  in  from   each   end,  putting  down  occasionally  the  sums 

from   the   beginning.     As   the  median  is  approached   put  them  all 

*  These  rough  and  ready  methods  of  estimating  the  probably  most  frequent 
measure  serve  for  any  studies  likely  to  be  made  by  the  non-mathematical  student. 
A  convenient  account  of  a  more  precise  method  will  be  found  in  the  Journal  of  the 
Royal  Statistical  Society  for  1896,  pp.  343-346. 


METHODS  OF  CALCULATION. 


75 


down.  The  median  will  then  fall  among  the  cases  of  some  one 
measure,  X  (Case  I.)  or  exactly  between  two  measures,  X  and  Xr 
In  the  latter  case  the  measures  may  be  side  by  side  on  the  scale 
(Case  II.)  or  separated  by  one  or  more  measures  the  frequency  of 
which  is  zero  (Case  III.).  Case  I.  is,  of  course,  by  far  the  most 
common.     Examples  are  given  below  : 


Case  I. 

Quantity. 

Frequency. 

9  up 

to  10 

2 

10      ' 

'     11 

4  (6) 

11      ' 

'     12 

9  (15) 

12      ' 

'     13 

14  (29) 

13      ' 

'     14 

16 

14      ' 

'     15 

13  (27) 

15      ' 

'     16 

8  (14) 

16      ' 

'     17 

5 

17      < 

'     18 

1 

Case  II. 

Case  III. 

Frequency. 

Frequency. 

3 

1 

5  (8) 

2 

11  (19) 

3(6) 

17  (36) 

8  (14) 

16  (36) 

12  (20) 

7  (14) 

4  (8) 

1  (7) 

2 

4 

2 

2 

Median  =  13.4 


Median  =  13.0        Median  =  13.5 


In  Case  I.  take  the  percentage  of  the  cases  of  the  one  measure 
in  which  the  median  lies  needed  to  make  the  sum  from  the  beginning 
one  half  the  total  number  of  cases ;  add  this  to  the  low  limit  of  X 
or  subtract  it  from  the  upper  limit  of  X,  according  to  the  direction 
in  which  you  are  taking  the  sums  from  the  beginning,  and  the  result 
is  the  median.*  It  is  often  a  sufficiently  close  approximation  to 
take  simply  the  central  value  of  ^Y. 

In  Case  II.  take  the  upper  limit  of  X1  or  the  lower  limit  of  X2 
which  are  of  course  the  same  thing. 

In  Case  III.  take  the  amount  half-way  ^between  the  upper  limit 
of  X  and  the  lower  limit  of  X„. 


Determination  of  the  Average  Deviation  from  the  Average. 

The  A.  D.  from  the  approximate  average  is  the  sum  of  the  de- 
viations of  the  individual  measures  from  it  (regardless  of  signs) 
divided  by  the  number  of  cases.  This  sum  is  given  in  the  course 
of  the  calculation  of  the  average  by  our  method.  Id  the  illustration 
it  is  92  +  148,  or  240. 

A.  D.  from  App.  Av.  =  -2g\0-,  or  2.475.     The  step  being  5,  this 

*This  is  not  absolutely  exact  since  the  frequency  of  the  different  measures  X 
low  limit  +  A,  X  low  limit  -f-  .2,  etc.,  will  rarely  be  exactly  the  same,  but  it  is  suf- 
ficiently accurate  for  any  mental  measurements  the  student  will  encounter.  A  cor- 
rection is  possible  only  when  the  exact  form  of  the  distribution  is  known. 


76  MENTAL    AND   SOCIAL    MEASCIWMEXTS. 

is,  in  thousandths  of  a  second,  12.375  ;  in  seconds,  .0124.  This  is 
incorrect  (1)  in  that  the  13  measures  .14.">  to  .150  have  been  regarded 
as  all  at  0  distance  from  .1475,  whereas  they  would  really  deviate 
from  it  even  up  to  half  a  step.  Thus  our  A.  D.  is  too  small.  On 
the  other  hand,  (2)  our  figure  is  incorrect  in  that  the  measures  of 
each  group  are  regarded  as  centering  at  its  mid-point,  whereas  really 
there  would,  as  a  rule,  be  more  of  them  in  the  half  of  it  nearer  the 
average  than  in  the  other  half.  Thus  our  A.  D.  is  too  large.  Cor- 
rections can  be  made  for  both  of  these  errors,  but  in  practice  it  does 
well  enough  to  compute  variabilities  from  a  fine  grouping,  say  into 
at  least  15  groups,  and  then  neglect  the  very  small  errors  resulting, 
since  they  partially  counterbalance  each  other. 

Finally,  the  sum  of  the  deviations  from  the  actual  averages  will 
differ  from  the  sum  of  those  from  an  approximate  average.  It  is 
easy  to  correct  for  this.  In  the  illustration  the  40  deviations  below 
should  each  be  increased  .58  of  a  step,  the  44  above  each  decreased 
.58  of  a  step,  and  the  13  zero  deviations  be  changed  each  to  —.58. 
This  would  give  an  increase  of  9  x  .58  step.  This  would  alter  the 
A.  D.  to  .0123.  This  correction  too  may  be  neglected  if  the  ap- 
proximate average  is  chosen  within  one  step.  If  it  is  not,  it  is  often 
as  easy  to  recalculate  the  deviations  from  the  actual  average,  or  a 
point  very  near  it,  as  to  make  the  correction. 

These  three  errors  may  be  called  the  errors  of  neglect  of  near 
deviations,  of  coarse  grouping,  and  of  the  approximate  average. 

Determination  of  the  Standard  Deviation  from  the  Average. 

Obtain  the  sum  of  the  square  of  the  deviations  from  the  approx- 
imate average  or,  if  it  is  not  wTithin  one  step  of  the  actual  average, 
of  the  deviations  from  a  point  that  is.  Then  calculate  o  from  the 
formula  i/(2a?)/n,  the  sc's  equaling  the  deviations  from  the  point 
chosen.  The  corrections  for  the  errors  of  neglect  of  near  deviations, 
of  coarse  grouping,  and  of  the  approximate  average  may  be  left  un- 
corrected without  serious  inaccuracy,  as  in  the  case  of  the  A.  D. 
The  correction  for  the  last  is  to  subtract  d2,  d  equaling,  as  before, 
(2a;)/w  (algebraic). 

In  the  illustration  if  150,  that  is,  a  point  just  between  the  145- 
150  and  150—155  groups,  is  taken  as  the  point  from  which  to  get  an 
approximate  <r,  the  calculation  is  as  follows : 


METHODS  OF  CALCULATION. 

2X(5.5)2=   60.50 

3X(4.5)2=  60.75 
11X(3.5)2  =  134.75 
13X(2.5)2=  81.25 
llX(l-5)2=  24.75 
13  X  (  .5)2==     3.25 


77 


365.25 

7X(  -5)2  = 

1.75 

8X(l-5)2= 

18.00 

13X(2.5)2  = 

81.26 

8X(3.5)2  = 

98. 

1X(4.5)2  = 

20.25 

3X(5.5)2  = 

90.75 

3X(6.5)2  = 

126.75 

OX  (7.5)2  = 

0X(8.5)2  = 

1X(9.5)2  = 

90.25 

527.00 
365  +  527  =  892 

l/W=  V$3, 
l/9J2  =  3.083  or,  in  seconds,  .01516.     a  =  .01516. 

It  is  much  easier  to  take  as  an  arbitrary  step  one  half  the 
regular  step  in  cases  where  the  chosen  point  is  just  between  two 
groups.  We  then  have  whole  numbers  to  deal  with.  The  above 
would  become  : 


2X(H)2  = 

=  242 

3X(9)2  = 

=  243 

11X(7)2  = 

=  539 

13X(5)2  = 

=  275 

HX(3)2  = 

=  99 

13X(1)2  = 

=  13 

1411 

7X(1)2  = 

=  7 

8X(3)2  = 

=  72 

13X(5)2  = 

=  275 

8X(7)2  = 

=  392 

1X(9)2  = 

=  81 

3X(H)2  = 

=  363 

3X(13)2  = 

=  507 

0X(15)2  = 

0X(17)2  = 

1X(19)2  = 

=  361 

2058 

1411  +  2058 

=  3569 

v*W  = 

i  3678 

78  MENTAL    AND  SOCIAL  MEASUREMENTS. 

j  3(5. S  =  6.07,  or  the  step  in  this  case  being  |  instead  of  5  as 
before,  .01518  sec. 

The  object  of  calculating  the  variability  from  an  approximate 
average  is,  of  course,  to  save  the  multiplication,  addition  and  squar- 
ing of  long  numbers.  In  general,  it  may  be  said  of  mental,  social 
and  physiological  measurements  that  it  is  wise  to  save  labor  in  their 
calculation  so  as  to  expend  it  in  getting  more  -or  more  accurate 
measurements.  By  the  methods  given  here  calculations  can  be  made 
very  rapidly. 

The   Determination  of  the  P.  E.  from  the  Average. 

The  P.  E.  equals  the  amount  of  deviation  from  the  average  (re- 
gardless of  signs)  which  is  exceeded  by  exactly  50  per  cent,  of  the 
deviations  of  the  individual  measures.  To  obtain  it  directly,  ar- 
range these  deviations  in  the  order  of  magnitude  and  find  the  point 
reached  in  counting  off  half  of  them.  For  instance,  in  the  case  on 
page  72  the  deviations  from  .1504  are  : 

Between  0  and  .0004  in  one  direction  and  between  0  and  .0046  in  the  other  7 

"     .0004  and  .0054     "           "          "          "        .0046-.0096  21 

"     .0054  and  .0104     "           "          "          "        .0096-.0146  24 

"     .0104  and  .0154     "           "          "          "  13 

The  total  number  of  cases  being  97,  it  is  sure  that  the  P.  E.  is 
somewhere  between  .0054  and  .0154. 

If  the  measurements  were  on  a  finer  scale,  it  could  be  located 
more  accurately  and  still  be  sure. 

A.  So  also  if  the  average  fell  exactly  at  the  mid-point  of  a  group 
or  just  between  two  groups.  For  instance,  if  the  average  in  the 
present  case  were  .150,  the  deviations  would  rank 


Between   0      and 

.005 

20 

"       .005  and 

.010 

19 

"       .010  and 

.015 

26 

The  P.  E.  would  then  surely  be  between  .010  and  .015.  We 
could  also  assume  that  the  9  J  of  the  26  deviations  between  .010  and 
.015,  which  are  needed  to  bring  us  to  the  50  per  cent,  point,  will 
bring   us    approximately    9.5/26    of  the    distance*   from    .010  to 

*  Really  a  little  less,  because  of  the  greater  frequency  of  measures  near  the 
average  than  of  those  more  remote  from  it  within  the  groups .  135-.  140  and  .  160-.  165. 


METHODS  OF  CALCULATION.  79 

.015,  that  is,  to  .0118.  The  P.  E.  then  would  be  approximately 
.0118. 

B.  In  so  far  as  the  measurements  are  distributed  symmetrically 
about  the  average,  the  P.  E.  calculated  directly  will  be  the  same  as 
the  distance  from  the  average  reached  by  counting  off  in  either  direc- 
tion 25  per  cent,  of  N  (the  total  number  of  measures  in  the  dis- 
tribution). This  would  again  be  the  same  as  the  distance  from 
the  average  reached  by  counting  in  25  per  cent,  of  N  from  either 
extreme. 

A  and  B  give  two  ways  of  reaching  quickly  an  approximate  P.  E. 
The  P.  E.  calculated  from  the  mid-point  nearest  the  average  or  from 
the  point  between  two  groups  nearest  the  average  will  be  a  close 
approximation  to  the  P.  E.  from  the  actual  average.  Its  calculation 
as  in  A  is  easy. 

In  so  far  as  the  distribution  is  approximately  symmetrical  (and 
when  it  is  not,  any  single  measure  of  the  variability  should  be  re- 
placed by  two  —  one  of  the  variability  above,  the  other  of  the  varia- 
bility below),  half  the  distance  between  the  25  percentile  and  75 
percentile  gives  a  very  close  approximation  to  the  P.  E. 

Determination  of  Quartiles,  Octiles  and  Other  Percentile  Values. 

The  determination  of  these  measures  has  only  one  difficulty,  that 
of  allowing  for  the  form  of  the  distribution,  which  commonly  makes 
cases  within  any  group  more  frequent  near  the  average.  For 
instance,  if  we  wish  to  find  the  lower  octile  in  the  case  given  on 
page  79,  we  have  n  =  465,  ^  ?i  =  58.125,  and  up  to  measure  20, 
55  cases,  3.125  cases  more  will  bring  us  to  the  octile  point.  How 
far  will  they  bring  us  from  20  toward  22.  If  the  16  cases  above  20 
and  below  22  were  evenly  distributed,  if  20.1,  20.2,  20.3,  etc.,  were 
equally  frequent,  it  would  be  correct  to  take  3.125/20  of  2  as  the 
distance  above  20  to  be  traversed.  But  the  general  form  of  the 
distribution  tells  us  that  the  measures  near  the  mode  are  more  likely 
to  occur.  For  perfect  exactness  an  allowance  should  be  made.  If 
the  groups  into  which  the  distribution  is  divided  are  few  in  number 
this  allowance  is  of  some  importance,  but  when  the  division  is  into 
15  or  more  groups,  the  simple  percentage  method  will  be  sufficiently 
exact  to  determine  quartiles  and  exact  enough  to  determine  octiles 
for  any  use  to  which  they  will  probably  be  put. 


BO  MENTAL     VND  soclM.    Mh'AsriUJMKXTS. 

Determination  of  the  Average  Deviation  and  of  the  Standard  Devia- 
tion from  t/ic  Mcdin/i. 

The  method  is  identical  with  that  described  under  'Determina- 
tion of  A.  D.  and  of  o  from  the  Average,'  except  that  the  approxi- 
mate average  there  should  be  replaced  by  '  approximate  median  '  and 
that  the  d  (Act.  Av.  aud  App.  Av.)  should  be  replaced  by  d  (Act. 
Median  and  App.  Median).     The  d  will  here  be  calculated  directly. 

Determination  of  the  P.  E.  from  the  Median. 

The  P.  E.  may  be  obtained  directly,  but  for  approximately  sym- 
metrical distributions  the  B  method  on  page  79  is  accurate  enough 
and  much  quicker,  viz.,  count  in  from  the  low  end  until  25  per  cent. 
of  the  cases  *  are  covered.  Call  the  quantity  thus  reached  the  25 
percentile.  Do  likewise  from  the  high  end  to  obtain  the  75  per- 
centile.    P.  E.  =  approximately  J  (75  percentile  —  25  percentile). 

It  is  wise,  in  general,  to  also  present  the  values  75  percentile  — 
median  and  median  —  25  'percentile,  which  represent  the  variability 
below  separately  from  that  above  the  median.  If  there  is  a  constant 
difference  between  the  two  in  series  of  measures  of  any  one  sort,  both 
should  be  given  to  show  the  skewness  of  distribution. 

Determination  of  Various  Percentile  Valves. 

The  limits  about  the  median  needed  to  include  any  given  per- 
centage of  cases  can  be  found  in  the  same  way. 

Determination  of  the  Average  Deviation  and  Standard  Deviation 
from  the  Mode. 

The  method  is  identical  with  that  described  under  'Determina- 
tion of  A.  D.  and  of  a  from  the  average,'  except  that  the  '  Approxi- 
mate Average '  should  be  replaced  by  mode  and  that  no  correction 
is  needed,  the  formulae  being  simply  : 

A.  D.  from  mode  =  Ix/n, 

a  from  mode  =  i/yx2/n. 

*  H  the  sums  from  the  beginning  have  been  jotted  down  during  the  calculation 
of  the  median,  the  25  and  75  percentile  points  can  be  found  in  less  than  a  minute. 


METHODS  OF  CALCULATION.  81 

Determination  of  the  P.  E.  from  the  Mode. 
The  P.  E.  may  be  calculated  directly  with  little  labor,  if  an  in- 
tegral measure  is  taken  as  the  mode.     In  other  cases  follow  the  A 
method  of  approximation. 

Determination  of  Various  Percentile  Values  from  the  Mode. 

The  methods  already  given  suffice. 

When  variabilities  are  measured  from  the  average  of  a  skewed 
distribution  (the  mode  should,  in  the  majority  of  skewed  distribu- 
tions, be  used  instead)  the  variability  above  and  that  below  the  av- 
erage should  be  given  separately.  That  is,  the  distribution  should 
be  divided  into  the  cases  above  and  the  cases  below  the  central  ten- 
dency, c.  Call  these  na  and  nb.  Then  find  the  average  deviation  of 
the  na  group  from  c  and  also  the  average  deviation  of  the  nb  group 
from  c.  For  a  do  the  same.  Instead  of  the  P.  E.  get  such  values 
as,  half  the  cases  of  na  deviate  less  than  so  much  from  c ;  half  the 
cases  of  nb  deviate  less  than  so  much  from  c ;  one  fourth  of  the  cases 
of  na  deviate  less  than  so  much  from  c,  etc.  The  methods  of  approx- 
imation allowed  hitherto  may  be  used.  A  sample  calculation  is 
given  below. 

In  multimodal  distributions  the  variability  should  be  calculated 
separately  for  the  distributions  into  which  the  given  distribution 
should  be  analyzed. 


Quantity. 

Frequency. 

Sums  from  Beginning. 

21,  i.  e., 

20.5  to  21.52 

22 

5 

7 

23 

16 

23 

24 

40 

63 

25 

60 

123 

26 

92 

215 

27 

100 

315 

28 

120 

29 

96 

531 

30 

84 

435 

31 

80 

351 

32 

70 

271 

33 

62 

201 

34 

48 

139 

35 

36 

91 

36 

20 

55 

37 

14 

35 

38 

10 

21 

39 

6 

11 

40 

2 

5 

41 

2 

3 

42 

1 

82 


MENTAL   AND  social   MEASUREMENTS. 


n  =  315  +  L20  +  531,  i.  e.,  n  =  966, 
mode  =  28,  i.  c,  27.5  to  2XJ>, 
na  =  531  +  60,  i.  c,  na  =  591,     nb  =  315  +  60,  I  e.,  nb  =  375, 

in   =  295.5,     In,  =  187.5. 
Points  reached  in  counting  in  295.5  from  42  and  187.5  from-21  are 
31.5  _  [(24.5/80)  x  1]  and  25.5  +  [(64.5/92)  x  1]. 
These  are  31.2  and  26.2. 

I  a    are  less  than  3.2  distant  from  the  mode. 

-  a 

1«       a      a        u      ]^g         u  a  u 

—  h 

Problems. 
16.  Calculate  the  average  and  the  A.  D.  and  a  from  it  in  each 
of  the  following  cases  ;  also  the  median  and  25  and  75  percentiles. 
Obtain  results  accurate  within  .5  the  unit. 


Case  I. 

Case  II 

Case  III. 

Quantity,            Frequency. 
11.00                    2 
12.00                    1 

Quantity. 
140  up  to  144 
144       "      148 

Frequency, 
1 
1 

Quantity, 

3  up  to  4 

4  "      5 

Frequency 
1 

3 

13.00 

4 

148       "      152 

4 

5 

<( 

6 

1 

14.00 

9 

152      "      156 

7 

6 

(< 

7 

3 

15.00 

21 

156      "      160 

13 

7 

1 1 

8 

4 

16.00 

11 

160      "      164 

20 

8 

(< 

9 

4 

17.00 

6 

164      "      168 

22 

9 

it 

10 

10 

18.00 

1 

168      "      172 

15 

10 

a 

11 

13 

19.00 

1 

172      "      176 

5 

11 

tt 

12 

13 

176      "      180 

2 

12 

a 

13 

18 

180      "      184 

2 

13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
2-5 
26 

n 
(i 

n 
a 
a 
ti 
it 
n 
n 
a 
(i 
(( 

14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 

16 

9 

15 

20 

10 

6 

7 

3 

1 

2 

2 

2 

0 

2 

17.  In  each  of  the  following  cases,  calculate  the  average  and  the 
A.  D.,  a  and  P.  E.  from  it ;  the  median  and  the  A.  D.,  a  and  P.  E 
from  it.     Accuracy  to  .5  the  unit. 


METHODS   OF  CALCULATION.  83 

Case  I. 

Number  of  Frenuenrv 

^'s  marked.  iiequency. 

14  up  to  16  2 

16,  etc.  0 

18  2 

20  2 

22  6 

24  3 

26  10 

28  12 

30  17 

32  28 

34  16 

36  30 

38  25 

40  30 

42  22 

44  23 

46  23 

48  13 

50  11 

52  11  ' 

54  11 

56  2 

58  1 

60  4 

62  5 

64  0 

66  1 

68  0 

70  1 

72  0 

74  0 

76  0 

78  1 
18.  In  Case  II.  of  17,  what  reasons  are  there  for  supposing  that 
the  grouping  that  follows  is  truer  to  the  real  facts  than  are  the  actual 

reported  measures '?  Calculate  average  and  A.  D.  for  this  second 

grouping. 

96.2  up  to  96.6  1 

96.6,  etc.  1 

97.0  5 

97.4  7 

97.8  28 

98.2  33 

98.6  42 

99.0  19 

99.4  14 

99.8  2 

100.2  2 


Case  II. 

Temperature 
at  mouth. 

96.0  up  to  96.2 

Frequency. 
2 

96.2,  etc. 

0 

96.4 

0 

96.6 

0 

96.8 

0 

97.0 

3 

97.2 

2 

97.4 

3 

97.6 

4 

97.8 

3 

98.0 

25 

98.2 

20 

98.4 

13 

98.6 

28 

98.8 

14 

99.0 

15 

99.2 

4 

99.4 

7 

99.6 

7 

99.8 

1 

100.0 

1 

100.2 

1 

100.4 

1 

84  MENTAL    AND  social   MEASUREMENTS.  . 

1!».  In  each  of  the  following  cases,  determine  the  mode  and  the 
variability  of  the  distribution  around  it.  Calculate  also  the  average 
and  the  variability  around  it. 


Case 

I. 

Case 

II. 

Weight  of  A. lult 

Englishmen.  * 

Frequency. 

Different  Rates 
of  Interest. 

Quantity  (of 
money  loaned) 

'."libs. 

up 

to  100  lbs. 

2 

100  etc. 

26 

4.00 

1014 

110 

133 

4.25 

45 

120 

338 

4.375 

40 

130 

694 

4.50 

17232 

140 

1240 

4.75 

1711 

150 

1075 

5.00 

22987 

160 

881 

5.17 

21 

170 

492 

5.25 

242 

180 

304 

5.50 

3293 

190 

174 

5.75 

27 

200 

75 

6.00 

6955 

210 

62 

6.25 

52 

220 

33 

6.50 

158 

230 

10 

6.67 

1449 

240 

9 

6.75 

59 

250 

3 

7.00 

2263 

260 

1 

7.25 
7.50 
8.00 
8.50 

7 

306 

1585 

892 

20.  In  the  report  f  from  which  Case  II.  is  quoted  the  4.0  =  4  or 
less  and  the  8.5  =  8.5  or  more.  If  these  facts  had  been  announced 
in  the  problem,  which  measures  only  could  have  been  calculated  ? 

*  Roberts'  'Manual  of  Anthropometry'  is  the  source  of  these  figures. 
fNew  Zealand  Official  Year-Book,  1901,  p.  231. 


CHAPTER   VII. 

THE    TRANSMUTATION   OF    MEASURES   BY   RELATIVE    POSITION   INTO 
TERMS    OF    UNITS    OF    AMOUNT. 

If  a  group  of  individuals  are  ranged  in  order  according  to  the 
amounts  which  they  severally  possess  of  a  trait,  we  can,  even  when 
ignorant  of  what  the  amounts  are  for  each  and  all  of  the  individuals, 
assign  to  each  the  amount  of  his  deviation  from  the  average,  pro- 
vided the  form  of  the  group's  distribution  is  known. 

For  instance,  let  100  boys  rank  with  respect  to  scholarship  as 

shown  in  Table  XXIX.,  and  let  the  form   of  distribution  be  that  of 

Fig.  68. 

TABLE  XXIX. 

100  Boys  a,  b,  c,  etc.,  Ranked  by  Relative  Position. 

1     a  is  the  highest  ranking  boy. 

3  b,  c,   d  are  the  next  highest  ranking  and  are  indistinguishable. 
6     e,   f,  g,   h,   i,    j 

10  k,   1,  m,   n,   o,  p,  q,   r,   s,  t 

15  u,  v,  w,  x,  y,   z,  a,  b,   c,  d,   e,  f,    g,   h,   i 

17  j,  k,    I,   m,  n,   o,  p,   q,   r,  s,    t,  u,    v,  w,  x,  y,  z 

19  A,   B,    C,    D,    E,   F,  G,  H,    I,  J,  K,  L,  M,    N,  O,    P,  Q,  R,  S 

14  T,  V,  V,  W,   X,  Y,  Z,   a,  (1,  y,   6,  e,     f,   // 

8        0,      l,     K,      \     fl,     V,    f,     O 

4  7T,     P,    C,      7, 
3       V,    (p,   x 


Fig.  us. 


If  we  build  up  approximately  the  distribution  of  Fig.  68  by  a 
series  of  40  rectangles  of  equal  base,  the  result  is  Fig.  69.  Call 
the  low  extreme  A  and  the  length  of  base  of  each  of  the  rectangles 

85 


86 


MENTAL    AND  SOCIAL   M ICASUn EM ESTS. 


K.  Then  the  upper  extreme  is  at  A  +  40 A'.  The  approximate 
distribution  in  terms  of  these  units  is  given  in  Table  XXX.  The 
frequencies  may,  of  course,  be  reckoned  on  the  basis  of  any  arbitrary 
unit.      In  Table  XXX..  the  total  area  is  taken  to  be  1,680. 


TABLE 

XXX. 

Quantity. 

In 

jquency. 

(Juan 

tity.               I 

'requency. 

Quantity. 

Frequency 

,1  to  ,1       A" 

7 

.1       14  A' 

74.5 

A  +  27iT 

26 

,1  •    KtoA   ■  2K 

20.5 

" 

15 

72.5 

u 

28 

22.5 

.!           IK.  etc. 

23 

(t 

16 

70 

a 

29 

19.5 

"      $K 

44 

(« 

17 

66.5 

it 

30 

16.5 

"       4 

52.5 

c< 

18 

63.5 

it 

31 

14 

"      5 

60.5 

(i 

19 

•    60 

(( 

32 

11.5 

"       6 

67.5 

C( 

20 

56 

C  i 

33 

9.5 

"       7 

73.5 

(< 

21 

52 

a 

34 

7.5 

"      8 

77.5 

<< 

22 

47.5 

(( 

35 

5.5 

"      9 

SO 

a 

23 

42 

a 

36 

4 

"     10 

80 

(i 

24 

38 

(( 

37 

2.5 

"     11 

79.5 

a 

25 

34 

u 

38 

2 

"     12 

78.5 

a 

26 

30 

A  +  39  K  to 

.4  +  40iT 

.5 

"     13 

77 

The  highest  ranking  boy,  a,  who  was  the  top  1  per  cent,  of  the 
group,  will  in  our  figure  occupy  the  top  1  per  cent,  in  the  table,  the 
highest  16.8  of  the  frequencies.  His  ability  theu  is  from  A  +  40A* 
part  way  into  ^1  +  347v.     The  abilities  of  the  next  three,  b,  c  and 


A+5K 


A+40K 


Fig.  69. 


d,  will  occupy  the  next  50.4  of  the  frequencies  and  be  included  be- 
tween the  limits  A  +  34.7  JT  and  A  +  SOAK.     So  on  with  the  next 
six  and  the  rest.     The  limits  for  each  group  are  shown  in  Fig.  70. 
The  average  ability  of  each  group  may  be  calculated  roughly  * 

*  By  a  subdivision  of  the  surface  into  finer  rectangles  the  precision  of  these  av- 
erages could  have  been  increased. 


MEASUREMENT  RY  RELATIVE  POSITION.  87 

from  the  facts  obtained  in  this  way.  Thus  the  highest  boy,  being 
represented  by  0.5  (A  +  39  A),  2  (A  +  SS.5K),  2.5  (.4  +  37.5iT), 
4(A  +  36.5K),  5.5  (A  +  35.57T)  and  2.3  (A  +  34.5iv),  has  as  an 
average  A  +  36.5iT. 

A  table  can  thus  be  formed  as  follows  : 
Boy  a  has  as  his  ability  A  +  M.bK ; 
Boys  b,  c,  d,  have  as  their  ability  A  +  32. 2 A"; 
Boys  e,  f,  g,  h,  i,  j,  have  as  their  ability  A  -f  2S.0K ; 
Boys  k,  1,  ni,  n,  o,  p,  q,  r,  s,  t,  have  as  their  ability  A  +  23. 8A";  etc. 


Fig.  70. 

These  measures  can  further  be  turned  into  distances  from  the 
mode  or  median  or  average  of  the  distribution  instead  of  from  its 
lower  limit  A.  They  can  be  put  in  terms  of  any  measure  of  the 
variability  of  the  scheme,  or  of  any  part  of  it  instead  of  K.  For  the 
distribution  given  in  Table  XXX.  can  be  used  in  every  way  like 
one  with  known  quantities  in  place  of  the  A  and  K.  For  instance, 
the  best  boy  is  +  2QK  from  the  mode,  or,  in  units  of  the  75  per- 
centile —  mode  measure  of  variability,  is  +  3.38. 

The  scholarship  of  every  boy  in  the  group  is  thus  represented  in 
definite  quantities  of  some  unit  of  amount  of  difference  from  some 
standard.  This  unit  itself  is  definable  as  the  difference  between  this 
person  and  that  person.  The  standard  is  similarly  definable  as  the 
scholarship  of  such  and  such  a  person. 

By  this  method  the  obscurest  and  most  complex  traits,  such  as 
morality,  enthusiasm,  eminence,  efficiency,  courage,  legal  ability,  in- 
ventiveness, etc.,  can  be  made  material  for  ordinary  statistical  pro- 
cedure, the  one  condition  being  that  the  general  form  of  distribution 
of  the  trait  in  question  be  approximately  known. 


MENTAL   AND  social  MEASUREMENTS. 

If  now  one  has  a  group  of  individuals  ranked  by  their  relative 
position  in  the  group,  his  first  task  before  he  can  transmute  the 
Beries  oi'  relative  positions  into  a  series  in  units  of  amount  is  to  ascer- 
tain the  form  of  distribution.  This  may  be  done  (1)  by  measuring 
objectively  in  units  of  amount  enough  sample  individuals,  or  (2)  if 
the  trait  cannot  be  measured  in  units  of  amount,  by  inferring  the 
form  of  distribution  from  that  of  similar  traits  which  can  be. 

1.  Suppose  one  had  2,000  ten-year-old  boys  measured  with  respect 
to  intellect  by  relative  position.*  If*  now  one  measured  200  of  them 
objectively  with  tests  scorable  in  units  of  amount,  he  could  properly 
transmute  the  2,000  on  the  basis  of  the  type  of  distribution  of  the 
200. 

2.  Suppose  one  had  1,000  individuals  measured  with  respect  to 
delicacy  of  discrimination  of  sound  by  relative  position.  (It  is  well- 
nigh  impossible  to  measure  sensitiveness  to  sound  in  objective  units 
which  another  observer  can  duplicate,  because  of  the  influence  of  size 
of  room,  resonance,  etc.)  It  is  fairly  certain  from  studies  of  the  del- 
icacy of  discrimination  of  length,  weight,  etc.,  that  delicacy  of  dis- 
crimination of  sound  is  distributed  in  something  approximating 
sufficiently  to  a  probability  surface,  with  range  of  from  +  3<r  to  —  3<r, 
to  prevent  calculations  on  that  basis  from  being  more  than  a 
little  wrong  on  the  average.  We  may,  therefore,  transmute  the 
1,000  measures  by  relative  position  into  units  of  amount,  on  the 
hypothesis  that  such  is  the  form  of  distribution.  So  also  with 
school  marks  if  intellect  in  general  is  found  to  follow  the  probability 
type  of  distribution. 

The  labor  of  transmutation  for  cases  which  follow  the  probability 
type  of  distribution  is  rendered  almost  nil  by  the  use  of  tables. 

If  the  probability  surface  of  range  +  3<r  to  —  3<r  is  divided  up  into 
100  equal  areas  representing  the  100  successive  per  cents,  from  the 
highest  to  the  lowest  of  the  total  group,  and  the  average  distance  from 
the  average  in  terms  of  a  is  calculated  for  each  per  cent.,  the  result 
is  Table  XXXI. 

If  now  we  ask,  '  What  will  be  the  average  ability  of  the  highest 
6  per  cent.? '  we  have  only  to  add  the  figures  for  the  first  6  per  cents, 
and  divide  by  6  (the  result  being,  of  course,  1.99).     Similarly  to  get 

*  Such  measures,  at  least  approximately  correct,  would  in  fact  be  easy  to  obtain 
through  school  marks,  teachers'  opinions,  personal  conferences,  etc. 


MEASUREMENT  RY  RELATIVE  POSITION. 


89 


TABLE  XXXI. 

Values,  in  Terms  of  the  Standard  Deviation  <t,  of  each  Single  Per  Cent., 
the  Distribution  Being  Normal.     Beginning  with  the  Extreme. 


'er  cents,  in  order  from 

highest  value  to  mode 

or  from  lowest  value 

to  mode. 

Value  in 
terms  of  <r. 

1st 

2.7 

2d 

2.18 

3d 

1.96 

4th 

1.81 

5th 

1.695 

6th 

1.598 

7th 

1.514 

8th 

1.439 

9th 

1.372 

10th 

1.311 

11th 

1.250 

12th 

1.200 

13th 

1.150 

14th 

1.103 

15th 

1.058 

16th 

1.015 

17th 

.974 

18th 

.935 

19th 

.896 

20th 

.860 

21st 

.824 

22d 

.789 

23d 

.755 

24th 

.722 

25th 

.690 

Per  cents,  in  order  from 

highest  value  to  mode 

or  from  lowest  value 

to  mode. 

Value  in 
terms  of  <r. 

26th 

.659 

27th 

.628 

28th 

.598 

29th 

.568 

30th 

.539 

31st 

.510 

32d 

.482 

33d 

.454 

34th 

.426 

35th 

.399 

36th 

.372 

37th 

.345 

38th 

.319 

39th 

.293 

40th 

.266 

41st 

.240 

42d 

.210 

43d 

.189 

44th 

.164 

45th 

.139 

46th 

.113 

47th 

.087 

48th 

.063 

49th 

.037 

50th 

.013 

the  average  ability  of  any  consecutive  series  of  per  cents.  Table 
XXXII.  gives  the  results  of  such  computation  for  every  consecu- 
tive series  in  the  upper  half  of  the  total  group.  If  the  signs  are 
changed  to  minus  it  serves  for  the  lower  half. 

The  figures  along  the  top  stand  each  for  the  per  cent,  already 
made  up  in  counting  in  from  the  extremes.  The  figures  down  the 
side  stand  for  the  per  cent,  in  the  group  for  which  a  measure  in 
terms  of  amount  is  to  be  found.  The  entries  in  the  body  of  the 
table  stand  for  the  average  amount,  in  terms  of  a,  of  any  per  cent, 
counted  in  from  any  point  to  the  average.  When  any  per  cent, 
passes  the  average  (e.  g.,  30  per  cent.,  often  40  per  cent.,  have  been 
used  up  in  counting  in  from  the  top)  it  is  necessary  to  take  from  the 
table  two  entries,  one  for  the   plus  cases  down  to  the  average,  the 


90  MENTAL   AND  social   MEASUREMENTS. 

other  for  the  minus  rases,  up  to  the  average,  of  which  the  percent,  is 
made  ap,  and  from  these  two  entries  to  compute  the  average  for  the 
given  p.r  cent.  Thus,  40  per  cent,  from  the  upper  extreme  having 
been  used  up,  the  next  30  per  cent,  will  average 

(+.13xl0)+(-.26x  20) 

_____  _    or  -  .Id. 

Illustrations  of  the  simpler  usage  in  cases  not  passing  the  aver- 
age are  as  follows : 
_ 

The  first  1  per  cent,  of  a  group  averages  +  2.7 
The    "    8    "      "        "       "         average +  1.86 
The  9th  and  10th  "     "       "  "      +1.34 

Per  cents.  6,  7  and  8  from  the  bottom  "       —  1.57. 


MEASUREMENT  BY  RELATIVE  POSITION.  91 

TABLE  XXXII  (a). 


0 

1 

2 

3 

4 

5 

6 

7 

1 

270 

218 

196 

181 

170 

160 

151 

144 

2 

244 

207 

189 

175 

165 

156 

148 

141 

3 

228 

198 

182 

170 

160 

152 

144 

137 

4 

216 

191 

177 

165 

156 

148 

141 

134 

5 

210 

185 

172 

161 

152 

145 

138 

131 

6 

199 

179 

167 

157 

149 

141 

135 

129 

7 

192 

174 

163 

153 

145 

138 

132 

126 

8 

186 

170 

159 

150 

142 

.135 

128 

124 

9 

181 

165 

155 

147 

139 

133 

126 

121 

10 

176 

161 

151 

143 

136 

130 

124 

119 

11 

171 

158 

148 

140 

134 

127 

122 

116 

12 

167 

154 

145 

138 

131 

135 

119 

114 

13 

163 

151 

142 

135 

128 

122 

117 

112 

14 

159 

147 

139 

132 

126 

120 

115 

110 

15 

156 

144 

136 

129 

123 

118 

113 

108 

16 

152 

141 

134 

127 

121 

116 

111 

106 

17 

149 

139 

131  ' 

125 

119 

113 

109  . 

104 

18 

146 

136 

129 

122 

117 

111 

106 

102 

19 

143 

133 

126 

120 

114 

109 

105 

100 

20 

140 

131 

124 

118 

112 

107 

103 

98 

21 

137 

128 

121 

116 

110 

105 

101 

96 

22 

135 

126 

119 

113 

108 

103 

99 

95 

23 

132 

124 

117 

111 

106 

101 

97 

92 

24 

130 

121 

115 

109 

104 

100 

95 

91 

25 

127 

119 

113 

107 

102 

98 

93 

89 

26 

125 

117 

111 

105 

101 

96 

92 

88 

27 

123 

115 

109 

104 

99 

94 

90 

86 

28 

120 

113 

107 

102 

97 

92 

88 

84 

29 

118 

111 

105 

100 

95 

91 

87 

83 

30 

116 

109 

103 

98 

93 

89 

85 

81 

31 

114 

107 

101 

96 

92 

87 

83 

79 

32 

112 

105 

99 

94 

90 

86 

82 

78 

33 

110 

103 

98 

93 

88 

84 

80 

76 

34 

108 

101 

96 

91 

86 

82 

79 

75 

35 

106 

99 

94 

89 

85 

81 

77 

73 

36 

104 

97 

92 

88 

82 

80 

75 

72 

37 

102 

96 

91 

86 

82 

78 

74 

70 

38 

100 

94 

89 

84 

80 

76 

72 

69 

39 

98 

92 

87 

83 

79 

75 

71 

67 

40 

97 

91 

86 

81 

77 

73 

69 

66 

41 

95 

89 

84 

80 

75 

72 

68 

64 

42 

93 

87 

82 

78 

74 

70 

66 

63 

43 

91 

85 

81 

76 

72 

69 

65 

62 

44 

90 

84 

79 

75 

71 

67 

64 

45 

88 

82 

78 

73 

69 

66 

46 

86 

81 

76 

72 

68 

47 

85 

79 

75 

70 

48 

83 

78 

73 

49 

81 

76 

50 

80 

92  MlStAI.    ASP   SOCIAL    MKASl'nEMKXTS. 

TABLE   XXXII  (b). 


8 

9 

10 

11 

12 

13 

14 

15 

1 

L87 

181 

125 

120 

115 

110 

100 

102 

o 

184 

128 

122 

118 

112 

108 

104 

99 

3 

131 

125 

120 

115 

110 

107 

102 

97 

4 

128 

128 

118 

113 

108 

104 

100 

96 

5 

126 

L20 

115 

111 

106 

102 

98 

94 

6 

128 

118 

113 

108 

104 

100 

96 

92 

7 

121 

116 

111 

106 

102 

98 

94 

90 

8 

118 

113 

109 

104 

100 

96 

92 

88 

9 

116 

111 

106 

102 

98 

94 

90 

86 

10 

114 

109 

104 

100 

96 

92 

88 

85 

11 

111 

107 

102 

98 

94 

90 

87 

83 

12 

109 

105 

100 

96 

92 

89 

85 

81 

13 

107 

103 

99 

94 

91 

87 

83 

80 

14 

105 

101 

97 

93 

89 

85 

81 

78 

15 

103 

99 

95 

91 

87 

83 

80 

76 

16 

101 

97 

93 

89 

85 

82 

78 

75 

17 

99 

95 

91 

87 

84 

80 

77 

73 

18 

98 

93 

89 

86 

82 

78 

75 

72 

19 

96 

92 

88 

84 

80 

77 

73 

70 

20 

94 

90 

86 

82 

79 

75 

72 

69 

21 

92 

88 

84 

81 

77 

74 

70 

67 

22 

90 

87 

83 

79 

76 

72 

69 

66 

23 

89 

85 

81 

78 

74 

71 

67 

64 

24 

87 

83 

80 

76 

73 

69 

66 

63 

25 

85 

82 

78 

74 

71 

68 

64 

61 

26 

84 

80 

76 

73 

70 

66 

63 

60 

27 

82 

78 

75 

71 

68 

65 

62 

58 

28 

80 

77 

73 

70 

67 

63 

60 

57 

29 

79 

75 

72 

68 

65 

62 

59 

56 

30 

77 

74 

70 

67 

64 

60 

57 

54 

31 

76 

72 

69 

65 

62 

59 

56 

53 

32 

74 

71 

67 

64 

61 

58 

54 

51 

33 

73 

69 

66 

63 

59 

56 

53 

50 

34 

71 

68 

64 

61 

58 

55 

52 

49 

35 

70 

66 

63 

60 

56 

53 

50 

47 

36 

68 

65 

61 

58 

55 

52 

49 

37 

67 

63 

60 

57 

54 

51 

38 

65 

62 

59 

55 

52 

39 

64 

61 

57 

54 

40 

62 

59 

56 

41 

61 

58 

42 

60 

MEASUREMENT  BY  RELATIVE  POSITION.  93 

TABLE  XXXII  (e). 


16 

17 

18 

19 

20 

21 

22 

23 

1 

97 

94 

90 

86 

82 

79 

76 

72 

2 

95 

92 

88 

84 

81 

77 

74 

71 

3 

94 

90 

86 

82 

79 

76 

72 

69 

4 

92 

88 

84 

81 

77 

74 

71 

67 

5 

90 

86 

82 

79 

76 

72 

69 

66 

6 

88 

84 

81 

77 

74 

71 

68 

64 

7 

86 

83 

79 

76 

72 

69 

66 

63 

8 

84 

81 

77 

74 

71 

68 

64 

61 

9 

83 

79 

76 

73 

69 

66 

63 

60 

10 

81 

78 

74 

71 

68 

65 

62 

59 

11 

79 

76 

73 

69 

66 

63 

60 

57 

12 

78 

74 

71 

68 

65 

62 

59 

56 

13 

76 

73 

70 

66 

63 

60 

57 

54 

14 

75 

71 

68 

65 

62 

59 

56 

53 

15 

73 

70 

66 

63 

60 

57 

54 

51 

16 

71 

68 

65 

62 

59 

56 

53 

50 

17 

70 

67 

64 

60 

57 

54 

52 

49 

18 

68 

65 

62 

59 

56 

53 

50 

47 

19 

67 

64 

61 

58 

55 

52 

49 

46 

20 

65 

62 

59 

56 

53 

50 

47 

45 

21 

64 

60 

58 

55 

52 

49 

46 

43 

22 

62 

59 

56 

53 

50 

48 

45 

42 

23 

61 

58 

55 

52 

49 

46 

43 

41 

24 

60 

57 

54 

51 

48 

45 

42 

39 

25 

58 

55 

52 

49 

46 

43 

41 

38 

26 

57 

54 

51 

48 

45 

42 

39 

37 

27 

55 

52 

49 

46 

44 

41 

38 

35 

28 

54 

51 

48 

45 

42 

39 

37 

29 

53 

50 

47 

44 

41 

38 

30 

51 

48 

45 

42 

40 

31 

50 

47 

44 

41 

32 

48 

46 

43 

33 

47 

44 

34 

46 

TABLE  XXXII 

CO- 

32 

33 

34 

35 

se 

37 

38 

39 

1 

45 

43 

40 

37 

35 

32 

29 

27 

2 

44 

41 

39 

36 

33 

31 

28 

25 

3 

43 

40 

37 

35 

32 

29 

27 

24 

4 

41 

39 

36 

33 

31 

28 

25 

23 

5. 

40 

37 

35 

32 

29 

27 

24 

21 

6 

39 

36 

33 

31 

28 

25 

23 

20 

7 

37 

35 

32 

29 

27 

24 

21 

19 

8 

36 

33 

31 

28 

25 

23 

20 

18 

9 

35 

32 

29 

27 

24 

21 

19 

16 

10 

33 

31 

28 

25 

23 

20 

18 

15 

11 

32 

29 

27 

24 

22 

19 

16 

14 

12 

31 

28 

25 

23 

20 

18 

15 

13 

29 

27 

24 

22 

19 

16 

14 

28 

25 

23 

20 

18 

15 

27 

24 

22 

19 

16 

26 

23 

20 

17 

24 

22 

18 

23 

94 


MENTAL   AND  social    MEASUREMENTS. 


TABLE  XXXII  {(I). 


24 

25 

26 

27 

28 

29 

30 

31 

1 

69 

66 

G3 

GO 

.-.7 

54 

51 

48 

2 

67 

64 

61 

58 

55 

52 

50 

47 

3 

66 

63 

60 

57 

54 

51 

48 

45 

4 

64 

61 

58 

55 

52 

50 

47 

44 

5 

63 

60 

57 

54 

51 

48 

45 

43 

6 

Gl 

58 

55 

53 

50 

47 

44 

41 

7 

60 

57 

54 

51 

48 

45 

43 

40 

8 

58 

55 

52 

50 

47 

44 

41 

39 

9 

57 

54 

51 

48 

4G 

43 

40 

37 

10 

56 

53 

50 

47 

44 

41 

39 

36 

11 

54 

51 

48 

46 

43 

40 

37 

35 

12 

53 

50 

47 

44 

41 

39 

36 

33 

13 

51 

48 

46 

43 

40 

37 

35 

32 

14 

50 

47 

44 

42 

39 

36 

33 

31 

15 

49 

46 

43 

40 

37 

35 

32 

29 

16 

47 

44 

42 

39 

36 

33 

31 

28 

17 

46 

43 

40 

37 

35 

32 

29 

27 

18 

44 

42 

39 

36 

33 

31 

28 

26 

19 

43 

40 

38 

35 

32 

30 

27 

24 

20 

42 

39 

36 

34 

31 

28 

26 

21 

40 

38 

35 

32 

30 

27 

22 

39 

36 

34 

31 

28 

23 

38 

35 

32 

30 

24 

36 

34 

31 

25 

35 

32 

26 

34 

TABLE 

XXXII 

(/) 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

1 

24 

21 

19 

16 

14 

11 

09 

06 

04 

01 

2 

•23 

20 

18 

15 

13 

10 

08 

05 

03 

3 

21 

19 

16 

14 

11 

09 

06 

05 

4 

20 

18 

15 

13 

10 

08 

05 

5 

19 

16 

14 

11 

09 

06 

6 

18 

15 

13 

10 

08 

7 

1G 

14 

11 

09 

8 

15 

13 

10 

9 

14 

11 

10 

13 

With  the  aid  of  Table  XXXII.  one  can  turn  measurements  by 
relative  position  into  measurements  in  units  of  +  or  —  a  almost  as 
fast  as  one  can  read. 

For  instance,  of  800  schoolboys, 

8  per  cent,  received  a  mark  of  E 
20  per  cent.    "     "     Trr' 


38  per  cent 
24  per  cent. 

8  per  cent. 

2  per  cent. 


VG 
G 
F 
P 
U 


MEASUREMENT  BY  RELATIVE  POSITION.  95 

The  table  tells  us  at  once  that,  in  so  far  as  the  distribution  of  the 
ability  in  the  group  in  question  follows  the  type  of  distribution  de- 
scribed above, 

E—  +  1.86<t 
VG=+  .93  a 
G=  +  .08  a 
F=  —  .79  a 
P=  — 1.58  ff 
U=  —  2.44ff 

There  is  still  another  possibility  of  turning  measures  by  relative 
position  into  units  of  amount  and  so  making  them  available  for  com- 
mon scientific  usage.  In  certain  cases  it  may  be  justifiable  to  sup- 
pose that  the  least  noticeable  difference  is  a  constant  quantity  for  any 
one  trait  for  any  one  observer ;  in  simpler  words,  that  if  I  say  that 
John,  James  and  Peter  are  to  me  indistinguishable,  say,  in  literary 
merit,  but  that  Henry  and  William  are  a  shade  better,  and  that 
George  and  Fred  are  a  shade  better  than  Henry  and  William,  the 
actual  difference  between  JJP  and  HW  equals  that  between  HW 
and  GF.  In  so  far  as  this  were  true  we  could,  with  a  large  group 
of  individuals  varying  continuously  from  the  low  to  the  high  extreme, 
make  groups  on  the  basis  of  the  least  noticeable  difference  and  call 
the  steps  of  ability  from  group  to  group  always  the  same. 

The  measures  are  then  identical  in  form  with  those  by  ordinary 
units  of  amount.  The  only  difference  is  that  the  amount  of  the 
quantity  at  the  starting-point  of  the  whole  group  (A)  and  the  amount 
of  the  step  from  one  subgroup  to  the  next  (A")  are  unknown  except 
from  the  things  measured  themselves  and  are  undefinable  except  in 
terms  of  them.  The  question,  'How  much  are  A  and  K1}  can  be 
answered  only  by  pointing  to  the  achievements  of  the  lowest  group 
and  saying,  '  That  is  A  J  by  pointing  to  the  differences  between  that 
group  and  others  and  saying,  '  This  much  difference  is  K,  this  much 
4K,  this  much  20 K  and  so  on.' 

The  hypothesis  that  the  least  noticeable  difference  in  a  trait  is 
for  the  same  observer  a  constant  quantity  has  not  been  tested  suffi- 
ciently to  decide  how  far  its  use  is  justifiable,  but  there  can  be  no 
doubt  that  some  modification  of  the  principle  involved  will  sometime 
be  a  valuable  resource  of  the  theory  of  mental  measurements. 

For  the  sake  of  simplicity,  only  the  case  of  individuals  measured 
by  their  relative  position  in  a  group  has  been  discussed  in  this  chap- 


96  MENTAL    AND  SOCIAL   MEASUREMENTS. 

ter.     Everything  in  the  chapter  applies  equally  well  to  measures  of 
the  different  trials  of  one  individual. 

Problems. 

21.  Turn  into  statements  in  units  of  the  A.  D.  of  the  distribu- 
tion, measured  -f  and  —  from  the  average,  the  measures  by  relative 
position  given  below;  first,  on  the  supposition  that  the  form  of  dis- 
tribution is  a  rectangle;  second,  on  the  supposition  that  the  form  of 
distribution  is  of  the  normal  type  (use  Table  XXXII.)  ;  third,  with 
no  supposition  about  the  form  of  distribution,  but  on  the  hypothesis 
that  the  measures  represent  a  grouping  by  the  least  noticeable  differ- 
ences and  that  these  differences  are  equal  : 

A,  B,  C,  D,  E  and  i^are  marks  running  from  high  to  low.  Of 
some  200  and  over  high-school  students,  2  per  cent,  received  A,  22 
per  cent.  B,  44  per  cent.  C,  25  per  cent.  D,  6  per  cent.  E,  and  1 
per  cent.  F. 

22.  Which  supposition  is  the  more  likely  ?     Why  ? 

23.  Using  Table  XXXL,  calculate  the  measure  in  terms  of 
units  of  amount  (1)  of  the  highest  four  per  cent,  of  a  group  normally 
distributed  ;  (2)  of  the  six  per  cent,  just  above  the  mode  ;  (3)  of  the 
three  per  cent,  from  the  end  of  the  seventeenth  down,  i.  e.,  of  per 
cents.  18th,  19th  and  20th.  Verify  the  results  from  the  entries  for 
these  groups  in  Table  XXXII. 

24.  On  the  hypothesis  that  the  distribution  of  darkness  of  eyes 
is  normal,  use  Table  XXXII.,  and  transmute  into  terms  of  units  of 
amount  the  following  measures  by  relative  position  : 


Eye  Color. 

Per  Cents. 

of  Englishmen.  * 

Light  blue. 

2.9 

call 

3 

Blue.     Dark  blue. 

29.3 

u 

29 

Gray.     Blue-green. 

30.2 

a 

30 

Dark  gray.     Hazel. 

12.3 

a 

12 

Light  brown.     Brown. 

11.0 

a 

11 

Dark  brown. 

10.8 

a 

11 

Very  dark  brown. 

Black. 

3.6 

" 

4 

It  is  possible  to  use  the  table  for  a  finer  scale  than  to  a  single 
per  cent,  by  interpolating.     But  it  is  hardly  worth  while. 

*  From  Galton's  'Natural  Inheritance.' 


CHAPTER  VIII. 

THE    MEASUREMENT    OF    DIFFERENCES    AND    OF    CHANGES. 

The  chief  questions  that  concern  the  measurement  of  differ- 
ences in  the  mental  sciences  arise  in  the  case  of  comparisons  of 
groups  and  measurements  of  changes.  Instead  of  any  general  ab- 
stract treatment  of  the  measurement  of  differences,  therefore,  I  shall 
present  the  special  applications  of  it  to  these  two  problems.  Only  a 
very  brief  outline  of  the  problem  as  a  whole  will  be  given  as  an  in- 
troduction. 

The  difference  between  any  two  amounts  of  the  same  kind  of 
fact  may  be  measured.     The  amounts  may  be  : 

1.  Two  single  figures,  each  standing  for  a  general  tendency,  e.g., 
averages,  medians  or  modes. 

2.  Two  single  figures,  each  standing  for  a  variability,  e.  g.,  A. 
D.'s,  <r's  or  P.  E.'s. 

3.  Two  single  figures,  each  standing  for  a  difference  itself. 

4.  Two  single  figures,  each  standing  for  a  relationship. 

5.  Two  total  distributions,  each  standing  for  a  general  tendency 
plus  the  deviations  from  it. 

The  general  tendency  may  be  to  the  possession  of  a  certain 
amount  of  variability,  of  difference  or  of  relationship,  as  well  as  of  a 
thing  or  quality.     It  will,  however,  commonly  be  the  latter. 

The  classification  above  could,  of  course,  be  extended  ad  infinitum 
with  such  complexities  as :  "  The  measurement  of  the  difference  be- 
tween two  variabilities,  each  being  of  the  amounts  of  relationship 
between  the  amount  of  difference  between  (1)  10-year-olds  and  11- 
year-olds  in  motor  ability  and  (2)  10-  and  11-year-olds  in  sense 
discrimination." 

The  difference  between  two  single  figures  will  be  measured  (a) 
by  the  gross  difference  ;  (6)  by  the  per  cent,  the  difference  is  of  the 
amount  of  one  of  them. 

The  difference  between  two  total  distributions  will  be  measured 
fully  by  comparing  them  item  by  item;  the  measurement  may  be 
summarized  in  various  ways. 

7  97 


MENTAL   AND  SOCIAL  MEASUREMENTS. 

The  difference  between  two  facts,  each  of  -which  is  measured  by- 
its  relative  position  in  a  series,  may  be  measured  most  satisfactorily 
by  transmuting  the  scries  and  then  using  regular  methods,  most 
quickly  by  the  gross  or  perceutile  difference  between  the  two,  rated 
as  members  of  the  same  series. 

The   Comparison   of  Groups. 

The  common  custom  of  comparing  groups  by  comparing  their 
averages  is  inadequate  because  for  both  practical  and  theoretical 
purposes  the  meaning  of  a  difference  between  two  averages  depends 
upon  the  variabilities  of  the  groups.  The  mere  fact,  for  example, 
that  in  the  A  test  (see  page  46)  the  averages  for  12-year-old  boys 
and  for  12-year-old  girls  were  respectively  41  and  46,  might  mean 

(1)  that  the  lowest  ranking  girl  was  above  the  highest  ranking  boy, 
i.  e.,  that  boys  and  girls  were  in  this  trait  totally  distinct  species  or 

(2)  that  only  5  per  cent,  of  girls  were  better  than  the  highest  rank- 
ing boy,  or  even  (3)  that  no  girl  was  equal  to  the  highest  ranking 
boy.  It  might  mean,  in  fact,  all  sorts  of  conditions,  some  of  which 
are  pictured  in  Figs.  71  to  76. 

It  is  of  no  great  advantage  to  estimate  the  difference  in  a  per 
cent,  rather  than  a  gross  amount.  One  group  may  in  ten  different 
tests  have  always  an  average  twenty  per  cent,  higher  than  the  other, 
and  yet  the  differences  in  ability  may  really  be  equal  in  no  two  of 
the  ten  cases.  For,  since  in  mental  and  social  traits  there  are  rarely 
absolute  zero  points  at  which  to  start  the  scale,*  the  meaning  of  each 
percentage  will  depend  upon  the  number  chosen  as  the  starting-point 
in  measuring.  We  can  always  make  a  difference  so  expressed  seem 
less  by  starting  the  scale  at  10  or  40  or  100  instead  of  at  0  or  4  or 
10.  And  the  same  percentage  in  a  case  where  the  variability  of  the 
trait  is  great  will  always  mean  for  practical  purposes  a  less  difference 
than  it  does  in  a  case  where  the  variability  is  small. 

For  instance,  if  the  A  test  is  scored  by  the  number  of  A's 
marked,  the  percentage  superiority  of  girls  to  boys  is  12.2  ;  if  by  the 
number  marked  more  than  the  lowest  12-year-old  record,  it  is  18.5 ; 
if  by  the  number  of  A's  omitted,  it  is  8.5.  Clearly  the  figure  de- 
pends on  an  entirely  arbitrary  factor. 

What  is  needed  for  the  comparison  of  groups  is  some  measure 

*See  Chapter  II.,  pp.  15  and  16. 


MEASUREMENT  OF  DIFFERENCES. 


99 


Figs.  71-76.  —  Graphic  comparisons  of  six  pairs,  the  difference  between  the 
averages  being  in  all  cases  the  same. 

which  (1)  will  inform  us  of  the  extent  to  which  the  two  groups  are 
separate  species,  the  extent,  therefore,  to  which  treatment  adequate 
for  one  group  will  be  inapplicable  to  the  other  and  which  (2)  will  be, 
so  far  as  is  possible,  commensurate  with  similar  measures  for  the  same 
groups  in  other  traits,  so  that  we  may  compare  the  differences  of 
groups  in  different  traits. 

The  first  desideratum  is  met  by  comparing  the  two  total  distri- 
butions instead  of  the  mere  averages,  or  approximately  in  the  case  of 
traits  somewhat  normally  distributed,  by  stating  the  variabilities  of 
the  two  groups.  Thus,  to  use  our  previous  illustration,  the:  distribu- 
tion of  12-year-old  boys  and  of  12-year-old  girls  in  the  A  test  as 
given  in  Table  XXXIII.  and  Fig.  77,  tells  us  at  once  that  the 
difference  between  the  averages  is  5.2,  that  over  99  per  cent,  of  the 
girls  are  contained  between  the  same  limits  of  ability  as  the  boys,  that 
only  31  per  cent,  of  boys  reach  the  median   mark   for  girls,  that  the 


It")  MENTAL  AXD  SOCIAL  MEASUREMENTS. 

sex  difference  is  far  less  important  practically  than  individual  differ- 
ences within  either  sex,  that  between  28  and  62  are  88.7  per  cent  of 
the  boys  and  87.4  per  cent,  of  the  girls.  These  same  measures  could 
be  obtained  approximately  from  the  theoretical  properties  of  the  nor- 
mal surface  of  frequency  if  the  variabilities  of  the  groups  were  given 
instead  oi^  the  total  distributions. 

The  second  desideratum  is  met  by  measuring  the  difference  in 
terms  of  the  per  cent,  of  one  group  who  reach  or  exceed  the  median 
mark  for  the  other  group  (or  some  other  set  measure).      If  in  Latin, 


TABLE  XXXIII. 

A' a 

Marked  in 

60  Seconds. 

Quantity. 

Frequency. 

12-year-old  boys.                   12 

-year-old  girls. 

14- 

16 

1 

16- 

18 

2 

18 

1 

1 

20 

2 

4 

2 

4 

4 

1 

6 

3 

2 

8 

9 

1 

30 

10 

2 

2 

8 

4 

4 

10 

11 

6 

15 

5 

8 

15 

9 

40 

10 

11 

2 

13 

9 

4 

12 

14 

6 

13 

10 

8 

8 

7 

50 

4 

6 

2 

6 

7 

4 

3 

6 

6 

2 

4 

8 

1 

8 

60 

4 

4 

2 

1 

4 

4 

1 

3 

6 

1 

8 

1 

70 

2 

1 

4 

1 

6 

MEASUREMENT  OF  DIFFERENCES.  101 

Greek,  algebra  and  history  one  group  of  students  always  show  30 
per  cent.,  reaching  the  median  of  another  group,  then  it  is  true  to  say 
that  the  second  group  is  equally  superior  in  all  four  of  these  studies. 
At  least  there  can  be  no  better  evidence  of  equality  in  amount  of 
difference  in  mental  traits  than  this. 

Under  the  present  conditions  of  thoughtless  measurements  of 
mental  traits  it  frequently  happens  that  groups  will  be  compared 
with  respect  to  the  same  trait  by  different  tests,  and  no  one  will  be  able 


14    18  SO  46  62  78 

Fig.  77.  — The  continuous  line  gives  the  distribution  of  ability  in  perception 
( A  test)  in  12-year-old  boys  ;  the  dotted  line  that  for  girls.  The  cases  are  grouped 
more  coarsely  than  in  the  table. 

to  tell  how  far  results  agree.  If  the  mere  averages  were  replaced  by 
the  measure  per  cent,  of  group  1  reaching  median  of  group  2,  results 
by  all  sorts  of  methods  could  be  put  together.  It  is,  of  course,  true 
that  when  one  group  so  far  exceeds  another  that  its  lowest  score  is 
above  the  highest  score  of  the  other,  the  method  suggested  here  fails. 
Such  cases  are,  however,  extremely  rare  in  the  comparisons  of  groups 
characterized  by  differences  of  sex,  training,  age,  social  conditions, 
birth,  occupation,  locality,  etc.,  such  as  psychology,  education  and 
sociology  are  studying. 

In  these  cases  of  total  disparity  in  the  two  distributions,  the  re- 
sults from  different  tests  may  be  made  commensurate,  so  far  as  is 
possible,  by  expressing  the  differences  in  terms  of  the  variability  of 
one  of  the  two  groups. 

Comparison  by  the  per  cent,  of  one  group  that  exceed  the  median 
measure  of  some  other  group  has  the  further  advantage  of  being 
applicable  to  groups  measured  by  relative  position  only.  For 
instance,  if  one  knew  that  the  crimes  in  one  town  were  as  listed  in 
column  1,  and  those  of  a  second  town  as  listed  in  column  2,  he  could 
state  that  almost  59  per  cent,  of  the  first  town's  crimes  were  greater 


102  MENTAL   AND  social   MEASUREMENTS. 

than  the  median  crime  of  the  second,  could  thus  have  a  quantitative 
comparison  of  the  two  without  having  to  adopt  speculative  equiva- 
lents of  one  crime  in  terms  of  others. 


Offense. 

Frequency  in 

lirst  town. 

Frequency  in 
second  town. 

Peddling  without  a  license 

2 

3 

Failure  in  jury  duty 

4 

5 

1  tisturbing  the  peace 

9 

11 

Drunkenness 

23 

28 

Robbery 

30 

27 

Assault  and  robbery 

17 

11 

Arson 

8 

10 

Murder  in  second  degree 

5 

4 

Murder  in  first  degree 

1 

1 

Patricide 

1 

In  comparing  groups  with  respect  to  variability,  allowance  must 
be  made  for  the  fact  that  the  amount  of  the  central  tendency  influences 
the  size  of  the  a  or  A.  D.  or  P.  E.  that  is  obtained.  For  instance, 
22  individuals  added  for  40  seconds,  and  gave  a  group  score  of — 
Median,  9.0  ;  A.  D.,  2.18.  The  same  22  individuals  then  added  for 
80  seconds  and  gave  a  group  score  of — Median,  16.0 ;  A.  D.  3.41.  In 
a  final  test  for  120  seconds,  the  results  were — Median,  23.5  ;  A.  D., 
5.18.  These  figures  do  not  mean  that  the  real  variability  of  the 
group  doubled  within  a  few  minutes,  or  that  it  altered  at  all,  but  only 
that  the  gross  amount  of  the  variability  depends  upon  the  gross  amount 
of  the  measures  themselves  as  well  as  upon  the  real  variability.  The 
gross  amount  of  variability  in  the  length  of  the  line  drawn  by  a 
group  of  individuals  trying  to  equal  a  10-mm.  line  will  be  far  less 
than  the  gross  variation  of  their  attempts  to  equal  a  1,000-mm.  line, 
yet  the  real  variability  is  presumably  the  same. 

Just  how  much  allowance  to  make  it  is  difficult  to  decide.  Karl 
Pearson  has  proposed,  as  a  measure  of  variability  by  which  groups 
may  be  fairly  compared,  the  gross  variability  divided  by  the  average. 
By  this  figure,  which  we  may  call  the  Pearson  Coefficient  of  Vari- 
ability, we  should,  in  the  case  of  the  12-year-old  boys  and  girls  in 
the  A  test  (Boys,  Av.  40.7,  A.  D.,  8.1  ;  Girls,  Av.  45.9,  A.  D., 
8.5)  reverse  the  gross  difference,  the  girls  becoming  only  93  per 
cent,  as  variable  as  the  boys.  It  would  seem  to  the  author  more  in 
accord  with  both  theory  and  facts  to  use  the  gross  variability  divided 
by  the  square  root  of  the  average.  Any  such  comparison  is  mislead- 
ing if  there  are  no  real,  but  only  arbitrary,  zero  points. 


MEASUREMENT  OF  DIFFERENCES.  103 

Comparisons  of  groups  in  variability  are  of  two  sorts  :  (1)  Of  dif- 
ferent groups  with  respect  to  their  variabilities  in  the  same  trait.  (2) 
Of  the  same  group  with  respect  to  its  variabilities  in  different  traits. 

In  the  first  case  the  differences  between  the  averages  in  the  cases 
which  interest  the  student  are  commonly  not  very  great,  and  the  zero 
points,  though  arbitrary,  are  subject  to  not  very  great  fluctuations ; 
consequently  the  comparison  by  any  method  is  commonly  such  as  to 
reveal  any  marked  difference  in  variability  that  exists.  In  practice 
one  can  do  no  more  than  present  the  two  total  distributions  the  vari- 
abilities of  which  are. to  be  compared,  explain  what  zero  points  were 
taken  and  why,  and  calculate  for  the  reader  the  relation  of  the  group's 
variabilities  by  all  three  methods.  Often  it  is  best  simply  to  present 
the  gross  variability  and  leave  any  one  to  allow  for  differences  in  the 
amount  of  the  measures  themselves  as  he  sees  fit. 

The  second  case  will  only  rarely  be  an  important  object  of  study. 
This  is  fortunate,  since  here  the  differences  between  averages  may 
run  to  any  amount,  and  the  zero  points  for  some  of  the  traits  may  be 
subject  to  extreme  variations.  For  instance,  suppose  that  one  wished 
to  compare  the  variabilities  of  adult  men  in  salary,  morality,  health  and 
intellect.  The  average  of  the  first  may  be  600  ;  that  of  the  second, 
10  ;  that  of  the  third,  1,000,  and  that  of  the  fourth,  10,000,  accord- 
ing to  the  units  and  zero  points  chosen.  We  would  take  as  our  zero 
point  for  salary  $0.00  per  year,  but  some  men  are  actually  a  burden 
and  should  be  rated  as  minus.  The  absolute  zero  point,  then,  some 
one  may  put  at  the  point  of  the  man  whose  work  is  worth  nothing  to 
any  one  and  whose  care  costs  the  most.  So  also  morality  may  be 
reckoned  upward  from  the  lowest  clergyman  or  from  the  lowest  crim- 
inal. Again,  is  the  zero  point  for  health  that  of  one  who  just  keeps 
above  dying  for  a  moment,  or  that  of  the  sickest  one  found  in  the  group? 

In  practice  one  can  do  no  more  than  to  present  the  total  distribu- 
tions, explain  what  zero  points  were  taken  and  why,  and  use  proper 
logic  in  inferring  anything  about  the  relations  of  the  variabilities 
found. 

The  Measurement  of  Changes. 

By  a  change  in  anything  is  meant  the  difference  between  two 
conditions  of  it.  It  might  seem  that  the  problem  of  the  measure- 
ment of  changes  was  identical  with  that  of  measuring  differences,  and 
that  this  section   was  superfluous.     In  a  certain  sense  this  is  true. 


104  MENTAL   AND  SOCIAL  MEASUREMENTS. 

The  general  principles  of  previous  chapters  do  answer  the  special 
questions  of  this  chapter.  But  it  will  be  clearer,  and  in  the  end 
save  the  student's  time,  to  study  these  special  questions  separately, 
especially  since  in  studies  of  change  one  is  commonly  concerned 
with  a  number  of  successive  steps  of  difference,  and  is  trying  to 
measure,  not  a  single  alteration,  but  a  continuous  process  of  alteration. 

The  Measurement  of  a  Change  in  an  Individual. 

A  mere  series  of  averages  does  not  give  the  data  for  a  complete 
measurement  of  the  change.  The  averages  might  be  the  same  and 
yet  the  constancy  of  performance  of  the  individual  might  have 
altered.  Thus  the  average  values  of  a  stock  from  1890  to  1900 
might  be  alike  and  yet  it  might  have  changed  from  a  fluctuating  un- 
certainty in  1890,  with  say,  an  average  deviation  of  40,  to  a  steady 
assured  value  in  1900,  with  an  average  deviation  of  only  3.  The 
stock  in  1890  would  be  more  desirable  property  than  the  stock  in 
1900  from  the  point  of  view  of  one  moved  by  the  gambler's  instinct ; 
the  reverse  would  hold  for  a  steady-going  man  with  a  family  or  for 
a  conservative  bank.  To  measure  change  fully  one  needs  a  series  of 
total  distributions.  If  they  are  not  at  hand  one  must  be  sure  not  to 
pretend  to  measure  something  other  than  that  represented  by  the 
series  of  quantities  he  does  have. 

Inequalities  iu  units  are  more  likely  to  escape  attention  in  meas- 
urements of  change  than  anywhere  else.  Yet  it  is  just  in  such 
measurements  that  they  may  do  the  most  harm.  For  instance,  all 
statistics  with  which  I  am  acquainted  measure  the  change  in  the 
death-rates  from  various  diseases  by  series  of  figures,  each  giving  the 
proportion  of  deaths  to  cases  or  to  total  population  or  to  some  other 
standard,  as  in  the  following :  * 

In  1891,  22.5  per  cent,  of  those  having  diphtheria  died. 

"  189^   2s*  2  "  "  "  "  "             " 

"  1893,  23.3  "  "  "  " 

"  1894,  23.6  "  "  "  "  "             " 

"  1895,  20.4  "  "  "  "  "             " 

"  1896,  19.3  "  "  "  "  " 

"  1897,  17.0  "  "  " 

"  1898,  14.8  "  "  "  "  " 

"  1899,  14.2  "  "  "  "  "             " 

"  1900,  12.8  "  "  " 

*  'London  Statistics,'  Vol.  XII.,  p.  97  of  the  Medical  Officer's  Report. 


MEASUREMENT  OF  DIFFERENCES.  105 

But  such  figures  can  not  be  taken  at  their  face  value,  for  to  cure 
one  case  of  diphtheria  is  not  the  same  quantity  of  progress  as  to  cure 
another.  The  progress  of  medicine  and  hygiene  which  reduces  the 
death-rate  from  40  to  30  does  so  presumably  often  by  curing  the 
easiest  quarter  of  those  previously  uncured.  The  next  cases  will  be 
harder,  and  possibly  to  cure  the  last  1  per  cent,  of  the  40  would 
mean  more  advance  in  medicine  and  hygiene  than  was  needed  for  the 
curing  of  all  the  other  99. 

When  the  change  is  in  number  of  individuals  affected  or  number 
of  errors  made  or  number  of  tasks  done,  there  is  then  special  danger 


4  — 


12rl3  13-14  14-15  16-16  16-17 

Fig.  78.  —  The  heights  of  the  line  above  the  base  line  at  the  points  12-13. 
13-14,  14-15,  15-16,  16-17,  give  the  differences  between  the  average  height  at  12 
and  that  at  13,  the  difference  between  the  average  height  at  13  and  that  at  14,  etc., 
for  25  boys  measured  annually  for  five  years. 

in  neglecting  the  inequalities  among  the  units ;  for  the  change  will 
commonly  single  out  the  easiest  first. 

The  common  absence  of  zero  points  in  the  case  of  mental  measure- 
ments makes  it  unwise  to  express  changes  in  percentile  increments, 
and  definitely  unjustifiable  to  so  express  them  if  the  gross  amounts 
whence  the  percentages  are  derived  are  not  also  given.  If,  for  in- 
stance, I  am  informed  that  A's  reaction  time  improved  10  per  cent, 
per  year  from  6  to  12  years,  I  am  at  a  loss  to  tell  what  is  meant. 

In  comparing  two  (or  more)  individuals  with  respect  to  change 
one  may  use  gross  change,  percentile  change  or  change  in  terms  of 
the  variabilities  of  the  individuals,  provided  that  he  makes  it  clear 
which  he  is  using  and,  of  course,  treats  both  individuals  alike.  No 
one  method  is  the  correct  one ;  all  are  correct,  but  measure  different 
things.  4  to  5  equals  8  to  9  if  by  change  is  meant  amount  added  ; 
4  to  5  equals  8  to  10  if  one  means  proportion  added  ;  4  to  5  (the  A. 
D.  of  4  being  2)  equals  8  to  9.5  (the  A.  D.  of  8  being  3)  if  one 


106  MENTAL  AND  SOCIAL  MEASUREMENTS. 

means  distance  traversed  toward  the  extreme  ability  of  the  previous 
condition.  This  is  all  that  can  be  said  in  general.  Each  special 
case  may  oiler  reasons  for  preferring  one  method.  The  beginner  in 
statistical  work  may  well  use  all  three. 

The  MecLSwremmt  of  a  Change  in  a  Group. 

This  heading  is  ambiguous  in  that  it  may  be  taken  to  refer  :  (1) 
to  the  measurement  of  the  changes  undergone  by  a  series  of  individ- 
uals, or  (2)  to  the  change  undergone  by  some  measure  of  a  group. 
It  should  be  needless  to  say  that  the  two  questions  are  radically  dif- 
ferent, but  they  are  often  confused.  The  changes  in  stature  of  100 
boys  from  the  age  1 5  to  the  age  1 6  are  not  the  change  from  the  aver- 
age stature  of  the  group  100  boys  at  15  to  the  average  stature  of  the 
same  group  at  16  years.  The  first  fact,  the  total  fact  of  all  the  in- 
dividal  changes,  is  calculated  from  100  individual  measures  of  change, 
is  a  distribution  with  an  ascertainable  variability  and  in  all  respects 
stands  in  the  same  relation  to  individual  changes  as  does  the  distri- 
bution of  an  ability  in  a  "group  to  the  abilities  of  its  members.  The 
second  fact  is  calculated  as  the  difference  of  two  averages,  has  no 
known  variability,  is,  in  fact,  simply  a  partial  measure  of  difference 
between  two  groups.  If  our  argument  is  ever  to  return  to  individual 
changes,  the  first  sort  of  measure  must  be  used.  This  will  commonly 
be  the  case. 

For  an  example  take  the  case  of  the  change  in  stature  of  25  boys 
from  the  twelfth  to  the  seventeenth  year.*  If  we  try  to  infer  any- 
thing about  growth  from  the  change  in  average  stature,  we  have  only 
the  following  facts:  Average  stature  for  12,  13,  14,  15,  16  and  17 
year  old  boys,  142.6,  148.12,  154.92,  161.60,  167.64  and  170.76 
centimeters  respectively.  Yearly  differences,  -f-  5.52,  -f  6.8,  +  6.68, 
-j-  6.04  and  +  3.12  centimeters.  These  differences  are  shown  in 
Fig.  78. 

If,  on  the  other  hand,  we  preserve  the  individual  changes  in  our 
statement,  we  have  the  facts  of  Table  XXXIV. 

These  show  the  great  variability  in  growth  and  the  law  of  com- 
pensation that  '  boys  who  were  tall  at  12  years  grow  the  faster  during 
the  interval  12  to  13  and  13  to  14  ;  but  during  the  intervals  of  14 

*For  these  measurements  I  am  indebted  to  the  kindness  of  Professor  Franz 
Boas  and  Dr.  Clark  Wissler. 


MEASUREMENT  OF  DIFFERENCES.  107 

to  15  and  15  to  16  they  grow  slowly  ;  with  the  boys  of  short  stature 
at  12  the  rates  of  growth  are  exactly  the  reverse.'*  How  the  single 
yearly  differences  above  fail  to  represent  the  real  complexity  and  cor- 
relation of  the  facts  can  be  seen  by  comparing  Fig.  78  with  Fig.  79, 
which  shows  the  real  changes  of  the  25  individuals.  Fig.  80  brings 
out  more  clearly  the  inverse  relation  between  the  change  from  12  to 
14  and  that  from  14  to  16. 


TABLE 

XXXIV. 

Growth 

of  25  Boys 

FROM    THE 

12th  through 
Change. 

the  17th  Year. 

Stature  at  12. 

12-13. 

13-14. 

14-15. 

15-16.                  16-17. 

132 

5 

7 

10 

6                    4 

134 

5 

5 

7 

10                    3 

135 

5 

2 

6 

8                    8 

135 

6 

7 

10 

6                    4 

136 

4 

8 

8 

7                   2 

136 

7 

9 

5 

3                    2 

137 

4 

6 

8 

6                    4 

137 

5 

4 

8 

10                    5 

139 

4 

8 

7 

7                    2 

140 

5 

7 

10* 

6                    3 

142 

9 

7 

6 

3                   1 

142 

4 

5 

5 

10                   6 

143 

4 

5 

5 

8                    7 

144 

6 

11 

6 

3                   1 

145 

6 

5 

7 

8                    4 

146 

4 

4 

6 

10                   4 

146 

4 

7 

8 

3                   1 

146 

4 

5 

4 

11                     2 

146 

9 

11 

5 

2                    1 

147 

4 

7 

9 

5                    4 

147 

8 

10 

7 

3                    1 

149 

7 

13 

1 

5                   1 

151 

5 

7 

8 

3                   4 

152 

.  5 

4 

10 

5                   2 

158 

9 

6 

1 

3                    2 

For  the  measurement  of  change  in  a  group  (that  is,  of  all  the  in- 
dividual changes),  the  statistical  treatment  is,  as  suggested  above, 
simply  that  for  any  fact  in  a  group,  the  fact  here  being  an  amount  of 
change  instead  of  an  amount  of  a  thing  or  condition.  The  need  of  a 
statement  in  a  table  of  frequencies  and  the  use  of  average,  mode, 
median  and  the  various  measures  of  variability  —  in  fact,  the  entire 
theory  of  Chapters  III.  and  IV.  —  are  applicable  here. 

* 'The  Growth  of  Boys,'  by  Clark  Wissler,  American  Anthropologist  (New 
Series),  Vol.  5,  pp.  83  and  84. 


L08 


MENTAL   ASD  SOCIAL  MEASUREMENTS. 


For  the  measurement  of  change  from  one  condition  of  a  group  to 
another  the  statistical  treatment  is  simply  that  described  in  the  case 
of  the  measurement  of  difference. 


12-13 


13-14 


14-15 


15-16  16-17 

Fig.  79.  —  The  heights  of  the  five  points  A,  B,  C,  D,  E  of  each  line  measure  the 
yearly  differences  for  one  individual  as  did  the  line  of  Fig.  78  the  yearly  differences 
for  the  average  stature  of  the  group.  The  figure,  that  is,  presents  graphically  the 
facts  of  Table  XXXIV. 

Problems. 

25.  In  which  trait,  A  or  B,  is  there  the  greater  difference  between 
Group  I.  and  Group  II  ? 

Quantity  B.  Frequency. 

Group  I.        Group  II. 


Quantity  A. 

Frequency. 

Group  I. 

Group  II. 

39 

1 

1 

40 

1 

0 

41 

4 

1 

42 

11 

7 

43 

23 

16 

44 

25 

20 

45 

28 

22 

46 

28 

26 

47 

30 

30 

48 

20 

27 

49 

9 

18 

50 

3 

10 

51 

0 

5 

52 

1 

2 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 


1 

0 

3 

7 

13 

20 

22 

15 

3 

3 

1 

1 


10 
16 
10 


MEASUREMENT  OF  DIFFERENCES. 


109 


26.  Groups  III.  and  IV.  are  approximately  normally  distributed. 
Group  III.  has  Median  =  10  and  A.  D.  =  4  and  Group  IV.  has 
Median  =  12  and  A.  D.  =  3.  What  per  cent,  of  Group  IV.  will 
exceed  the  median  for  Group  III.  ?  What  per  cent,  of  Group  III. 
will  exceed  the  median  for  Group  IV.  ?  (The  table  on  page  60 
affords  the  further  data  necessary.) 


-20 


6 


12414 


14-16 


Fig.  80.  — The  height  of  any  one  of  the  lines  at  its  left-hand  extreme  measures 
the  change  in  stature  of  one  boy  from  12  to  14  ;  its  height  at  the  right  hand  extreme 
measures  the  change  from  14  to  16. 


27.  If  Ave  know  the  average  wealth  of  100  men  in  1900  to  be 
$5,000  and  in  1905  to  be  $10,000,  what  do  we  know  about  the 
changes  that  have  taken  place  ? 

28.  Recall  any  arguments  based  on  the  application  to  individuals 
of  some  change  true  of  them  only  as  a  group.  Where  else  have  we 
in  this  book  met  a  similar  fallacy  ? 


CHAPTER  IX. 

THE    MEASUREMENT    OF    RELATIONSHIPS. 

The  difficulty  of  measuring  mental  and  social  relationships  is,  of 
course,  due  to  their  variability.  The  relation  of  the  weight  of  a 
gas  at  constant  temperature  and  pressure  to  its  volume  we  assume 
to  be  always  the  same,  but  the  relation  of  intellect  to  morality  is 
almost  never  the  same ;  the  relation  of  the  force  of  gravity  to  the 
product  of  the  masses  of  the  two  bodies  is  constant,  but  the  relation 
of  ability  in  school  to  efficiency  in  life  is  very  variable.  The  prob- 
lem is  thus  to  represent  the  total  tendency  shown  by  many  different 
individual  relationships. 

Case  I. 

The  relationship  of  changes  in  the  amount  of  one  thing  to 
changes  in  the  amount  of  another  thing,  when  the  things  are  physical, 
is  shown  by  a  series  of  corresponding  values  of  the  two  things 
reckoned  from  zero  points  in  both  cases,  each  pair  of  values  being 
represented  by  two  constants.  It  is  expressed  mathematically  by 
the  equation  which  represents  the  way  in  which  the  amount  of  the 
one  thing  depends  upon  the  amount  of  the  other. 

The  following  case  may  serve  as  an  illustration  : 

n  =  the  index  of  refraction  of  air. 

d  =  the  density  of  air. 

/>  (a  quantity  subject  to  the  control  of  the  experimenter)  =  Cxd. 

N  (a  quantity  measurable  by  the  experimenter)  =  C2{n  —  1). 

Cj  and  C2  are  constants. 

The  experiments  consisted  in  varying  p  and  measuring  the  re- 
lated changes  in  N.     The  results  are  as  follows  : 

When  p  is    9.989  N  is    316.7 

"  "  10.146  "  "     321.2 

"  "  10.163  "  "     321.6 

"  "  18.281  "  "     579.2 

"  "  18.365  "  "     582.7 

"  "  26.932  "  "     852.6 

"  "  35.990  "  "  1142.1 

"  "  48.780  "  "  1545.1 

110 


MEASUREMENT  OF  RELATIONSHIPS. 


Ill 


If  each  of  these  pairs  of  related  values  is  turned  into  an  equation 
of  the  form  JV=  xp,  the  results  are  : 


^r=3l.70p 

N--=z\.mP 

N  =  3l.64p 

N=31.68p 


N=31.72p 
N  =  Sl.ffip 
N  =  S1.Q9P 
N  =  31.Q8p 


Obviously,  a  single  equation  N=  31.68/>  expresses  very  closely 
the  relationships  found  for  different  values  of  p. 

The  measurements  of  relationship  here  are,  of  course,  not  abso- 
lutely free  from  variability.  For  instance,  the  10.163  came  really 
from  7  measurements  with  an  average  deviation  of  .012.     But  the 


two 


WOO 


wo 
200 
100 
{CO 


Amounts  o£  P 


to 


10  30 

Fig.  81. 


HO 


SO 


112  MENTAL   AND  SOCIAL  MEASUREMENTS. 

variability  is  here  small  and  presumably  due  entirely  to  variations  in 
the  instruments  or  observers. 

It'  the  pairs  of  values  are  plotted  as  in  Fig.  81,  the  slope  of  the 
line  shows  the  relationship.  The  equation  N=  31.68/)  expresses 
very  closely  the  slope  of  this  line  referred  to  its  coordinates.  X  p 
is  thus  constant,  (n  —  l)/d  equals  Njp  times  some  constant.  There- 
fore, (»  —l)/d  itself  equals  a  constant.  The  relation  between  the  index 
of  refraction  of  air  and  its  density  is  then  such  that  (ii  —  l)/d  =  k  or 
n  =kd+  1* 

Case  II. 

When  changes  in  the  amount  of  a  mental  trait  are  to  be  related 
to  changes  in  the  amount  of  a  physical  trait,  the  series  will  be  of 
pairs,  of  which  one  will  be  a  constant  and  a  quantity  measured  from 
a  zero  point  and  the  other  a  variable  and  often  a  quantity  with  no 
ascertained  zero  point.  The  following  case  may  serve  as  an  illus- 
tration : 

Ebbinghaus  in  studying  the  relation  between  the  lapse  of  time 
and  memory  found  that  if  a  series  of  syllables  was  memorized  and 
then  24  hours  allowed  to  pass,  there  was  required  to  rememorize  the 
series  73.6  per  cent,  as  much  time  as  was  originally  needed.  In 
another  test,  however,  the  result  was  60.4  per  cent.,  and  he  quite 
properly  announces  not  only  the  average  of  all  the  numerous  varying 
results,  but  also  each  separate  one.  So  also  for  the  time  taken  after 
intervals  of  19,  63  and  525  minutes  and  2  and  6  days.  In  the 
statement  of  the  relationship  which  follows  (in  Table  XXXV.), 
the  '  time  saved  in  learning '  quite  evidently  is  a  variable.  One  may 
note  the  wisdom  of  the  investigator  in  measuring  the  change,  not 
in  the  ambiguous  units  of  so  many  words  lost,  but  in  '  per  cent,  of 
original  time  taken  to  relearn,'  a  system  of  units  with  an  intelligible 
zero  point. 

If  we  plot  the  pairs  of  values  as  in  the  previous  illustrations, 
the  result  is  Fig.  82,  which  shows  the  general  tendency  of  the 
relationship  and  at  the  same  time  its  lack  of  uniformity. 

In  such  cases  it  is  common  to  replace  the  tables  of  frequencies 
for  the  mental  trait  by  their  averages.     This  procedure  never  fully 

*  The  figures  in  this  illustration  are  quoted  from  a  report  by  Henry  G.  Gale  of 
a  research  'On  the  Relation  between  Density  and  Index  of  Refraction  of  Air.' 
Physical  Review,  January,  1902. 


MEASUREMENT  OF  RELATIONSHIPS.  113 

describes  the  relationship  and,  unless  the  distributions  are  sym- 
metrical about  a  central  mode,  may  misrepresent  it.  At  all  events, 
the  total  fact  of  the  relationship  should  always  be  presented,  as  well 
as  its  abbreviated  and  more  convenient  form.  In  so  far  as  the  zero 
point  from  which  the  mental  trait  is  measured  is  unknown,  it  is 
necessary  to  replace  all  face  values  y,  yv  y2  etc.,  of  the  mental 
traits  measured  by  k  +  y,  k  +  yv  k  +  y2,  etc.  The  formulation  of 
any  algebraic  expression  for  the  relationship  is  thus  less  simple. 

TABLE  XXXV. 
Relation  Between  Lapse  of  Time  and  Memory.* 

0.32  hrs.  1.05  hrs.         8.75  hrs.         24  hrs.  48  hrs.  144  hrs.  744  hrs. 


64.3 

49.6 

36.0 

26.4 

17.4 

21.0 

26.0 

20.0 

55.9 

37.4 

29.0 

39.6 

32.7 

31.1 

31.6 

19.4 

56.6 

47.4 

28.0 

35.4 

12.3 

32.7 

34.7 

22.9 

62.5 

46.8 

30.4 

39.9 

28.9 

24.4 

31.6 

6.7 

60.7 

51.4 

39.8 

34.9 

30.6 

17.7 

30.3 

6.9 

63.1 

49.1 

35.6 

38.9 

46.0 

5.9 

20.5 

25.9 

59.1 

44.5 

48.2 

46.7 

23.5 

34.1 

10.1 

18.9 

56.0 

54.5 

31.6 

16.7 

25.4 

33.3 

6.8 

20.5 

64.4 

42.3 

35.5 

21.3 

18.4 

28.7 

6.5 

11.4 

44.7 

40.9 

40.1 

38.6 

23.4 

23.2 

13.3 

17.3 

53.6 

34.2 

37.9 

29.0 

41.0 

40.3 

17.7 

17.1 

57.7 

45.4 

38.0 

37.8 

29.5 

37.9 

17.1 

32.8 

35.8 

36.5 

33.9 

26.5 

15.9 

31.4 

35.9 

29/7 

44.9 

20.1 

27.6 

16.4 

51.3 

37.0 

17.5 

39.7 

13.2 

36.2 

50.0 

14.9 

42.4 

2.5 

27.6 

13.4 

45.6 

6.4 

36.2 

23.6 

31.0 

30.1 

22.8 

5.3 

20.9 

7.9 

24.6 

31.6 

27.9 

24.8 

36.9 

37.0 

30.2 

19.0 

25.0 

14.1 

44.4 

19.7 

21.0 

25.2 

6.7 

45.8 

31.9 

31.4 

43.7 

16.7 

30.6 

14.8 

19.7 

23.7 

42.5 

32.3 

20.9 

19.8 

37.6 

24.4 

32.1 

26.7 

34.8 

Averages,  ■ . — 

58.2  44.2  35.8  33.7  27.8  25.4  21.1 

*  From  Herm.  Ebbingbaus,  '  Uber  das  Gediicbtniss,'  pp.  93-103. 


114 


MEXTAL   AXD  SOCIAL   MEASUREMENTS. 


60 

50 

y 

• 

40 

1 

CO 

X 

30 

20 

o 

10 

*5 

o 

M 

auti.5     »-* 

20      100 


Fig.  82. 


500 


Case  /J/. 

If  one  mental  trait  is  to  be  related  to  another  the  amounts  of  one 
are  treated  each  as  a  constant  and  the  problem  is  that  of  Case  II., 
except  for  the  fact  that  both  series  of  amounts  must  be,  unless  there 
are  real  zero  points,  expressed  as  k1  +  yv  hx  -f  y2,  \  -f  ys,  etc.,  and 
\  +  xv  lc2  +  x2,  etc. 

The  following  case  may  be  taken  as  an  illustration  : 

The  relationship  between  the  ability  to  perceive  A's  scattered 
among  other  capital  letters  and  the  ability  to  perceive  words  con- 
taining both  a  and  t  scattered  among  other  words,  the  ability  being 
measured  in  schoolgirls  all  of  the  7B  grammar  grade.  The  amounts 
to  be  related  are  the  number  of  A's  marked  in  60  seconds  and  the 
number  of  words  containing  a  and  t  marked  in  120  seconds. 

The  related  amounts  found  by  measurement  are  given  in  Table 
XXXVI.  The  zero  points  being  unknown,  these  pairs  should  all  be 
turned  into  \  +  10  with  \  +  36,  \  +  10  with  /;,  -f  51,  etc.  If 
each  related  pair  is  plotted  as  before,  our  ignorance  of  the  zero  points 
would  be  expressed  by  leaving  the  axes  of  reference  undetermined 
save  in  their  direction,  as  in  Fig.  83. 


MEASUREMENT  OF  RELATIONSHIPS. 


115 


TABLE 

XXXVI. 

a-t  words 

A's 

a-t  words 

A's 

a-t  words 

A's 

a-t  words 

A's 

marked. 

marked. 

marked. 

marked. 

marked. 

marked. 

marked. 

marked 

10 

36 

17 

47 

20 

58 

23 

62 

10 

51 

17 

49 

20 

60 

23 

65 

11 

43 

17 

57 

20 

61 

23 

70 

11 

47 

18 

41 

20 

62 

24 

55 

11 

56 

18 

43 

20 

64 

24 

55 

12 

45 

18 

46 

20 

76 

24 

59 

12 

46 

18 

47 

21 

45 

24 

78 

13 

52 

18 

47 

21 

46 

25 

49 

13 

55 

18 

51 

21 

47 

25 

54 

14 

48 

18 

51 

21 

48 

25 

59 

14 

58 

18 

53 

21 

49 

25 

70 

15 

37 

18 

62 

21 

50 

25 

78 

15 

38 

18 

62 

21 

54 

25 

81 

15 

42 

18 

63 

21 

54 

26 

57 

15 

43 

18 

66 

21 

57 

26 

60 

15 

47 

19 

57 

21 

59 

27 

61 

15 

50 

19 

60 

21 

59 

27 

64 

15 

52 

19 

61 

21 

61 

27 

65 

15 

64 

19 

64 

21 

63 

27 

67 

15 

72 

20 

38 

21 

65 

27 

74 

16 

43 

20 

43 

22 

47 

27 

78 

16 

46 

20 

45 

22 

48 

28 

54 

16 

46 

20 

46 

22 

53 

28 

65 

16 

55 

20 

48 

22 

59 

28 

65 

16 

56 

20 

50 

22 

62 

29 

69 

16 

67 

20 

51 

22 

62 

30 

49 

16 

70 

20 

52 

22 

63 

30 

59 

17 

39 

20 

56 

22 

77 

30 

81 

17 

42 

20 

56 

23 

45 

34 

73 

17 

44 

20 

56 

23 

48 

17 

45 

20 

57 

23 

58 

Fig.  83. 


L16  MENU  I  /.   .  I ND  SOCIAL  MEASUREM  EN  TS. 

Case  IV. 
The  difficulty  with  zero  points  could  be  overcome  if  no  attempt 
were  made  to  measure  the  relations  of  absolute  amounts,  but  only  the 
relations  of  excesses  or  deficiencies  similarly  measured  in  a  second  trait. 
Thus,  tor  instance,  one  may  ask  the  relationship  of  the  number  of 
.l's  marked  in  a  minute  more  than  10  to  the  number  of  a-t  words 
marked  in  two  minutes  more  than  4  ;  or  the  relationship  of  the  num- 
ber of  A's  marked  in  a  minute  by  ten-year-old  boys  more  than  the 
lowest  record  to  a  similar  measure  for  a-t  words  ;  or  a  similar  ques- 
tion with  the  average  performance  as  the  zero  point  in  both  cases. 
The  last  question  is  one  that  the  mental  sciences  often  ask ;  for  the 
mental  sciences  are  more  frequently  interested  in  the  relationship  of 
deviations  in  one  trait  from  the  general  type  to  deviations  in  some 
other  trait  again  from  the  general  type,  than  in  the  relationship  of 
gross  amounts  of  the  trait.     The  measurement  now  is  simply  of  the 

TABLE   XXXVII. 


a-t 
Fords. 

A's 

a-t 
words. 

^'s             words.          A'a 

a-t 
words. 

A's 

n-t 
words. 

^'s 

-10 

—  19 

—  3 

—  16 

0—7 

+  1 

+  10 

+    7 

4-  6 

—   4 

—  13 

—   5 

4-2 

—    8 

+    9 

—  9 

—  12 

—  11 

—   4 

—   7 

+  10 

—   8 

—  10 

—   3 

—   2 

+  12 

+    1 

—   8 

4-  1 

+    4 

4-19 

—  8 

—  10 

—   6 

+  1 

+    7 

+  23 

—  9 

+    2 

4-   1 

+    7 

4-    8 

—    1 

—  7 

—   3 

—  2 

—  14 

+    2 

4-  8 

+  10 

0 

—  12 

+    3 

+  22 

+  10 

—  6 

—  7 

—   9 

+    5 

4-3 

—  10 

4-  9 

+  14 

+    3 

—   8 

+    6 

—   7 

+  10 

—    6 

—  5 

—  18 

—   8 

4-  7 

+   3 

4-   4 

—  17 

—   4 

4-    9 

+    7 

+  26 

—  13 

—   4 

+  21 

+  10 

+  14 

+  18 

—  12 

—   2         J 

-1      —10 

+  15 

—  8 

+    7 

—   9 

4-4 

0 

—    5 

4-  7 

—    8 

0 

—   3 

+    8 

—   7 

+   4 

+    9 

+  11 

—   6 

+  23 

+  17 

—  1 

+    2 

—   5 

+  5 

—   6 

—  4 

—  12 

—  9 

—  9 

4-    5 
+    6 
+    9 

—  1 

—  1 

+    2 

—   1 

4-  4 

+  15 

0 

0 

—  17 

+   4 

+  23 

+    1 

—  12 

+    4 

+  26 

+  12 

—  10 

+    6 

+  6 

+    2 

+  13 

—   9 

4-  8 

+    5 

MEASUREMENT  OF  RELATIONSHIPS. 


117 


relationship  of  the  differences  -4-  °r  —  of  one  trait  from  its  typical 
condition  to  similar  differences  of  the  other  trait. 

If,  after  turning  each  of  the  measures  of  the  previous  illustration 
into  terms  of  so  much  +  or  —  the  central  tendency  of  the  series  to 
which  it  belongs,  we  treat  them  as  in  Case  III.,  we  have  the  results 
given  in  Table  XXXVII.  and  Fig.  84. 

These  results  represent  in  available  form  the  relationship  as 
found  in  each  of  the  122  cases  studied.  The  variation  among  them 
is  so  great  that  any  single  law  can  express  only  the  general  tendency, 
a  tendency  from  which  the  individuals  often  diverge  very  much. 


+15 
+10 

+  5 


■5 

10  r!  :         ',  1 

r-r>  * 

<  : 

A-T    Wcrds.~ 

-8  -4 


-16 


0  +  4 

Fig.   84. 


+12 


Each  case  of  the  relationship  is,  as  has  been  shown,  represented 
by  the  position  of  a  point  with  reference  to  two  axes  or  by  an  equa- 
tion, trait  1  =  some  function  of  trait  2,  A  =  F  of  B.  Such  tables 
and  figures  as  XXXVI.  and  XXXVII.  and  83  and  84,  express 
together  all  the  cases  of  a  relationship  which  one  has  measured. 
The  present  problem  is  to  find  some  simpler  means  of  presenting  the 
general  tendency  manifested  by  the  total  group  of  cases  of  the 
relationship. 

The  obviously  useful  habit  of  classifying  the  cases  according  to 
the  amounts  of  one  quantity  to  which  the  other  is  to  be  related  has 
already  been  adopted.  The  group  of  measures  in  trait  2  related  to 
any  single  measure  in  trait  1  is  called  the  array  correlated  with  that 
amount  of  1.  —  19  A'e  and  —  I  .  I's  form  the  array  correlated  with 
—  10  of  a-t  words;  —  12,  —  8  and  +  1  form  tlrt'  array  correlated 
with  —  9   <i-l   words,  etc.      If  in    place   of  each   array  one   takes   its 


US 


MENTAL    AND  SOCIAL  MEASUREMENTS. 


central  tendency,  the  complicated  table  becomes  the  simple  series  of 
equations  of  Table  X  X  X  Y  1 1 I. ;  the  complicated  diagram  the  simple 
series  of  point-  referred  to  two  axes  shown  in  Fig.  85.  The  problem 
is  reduced  to  the  same  problem  as  in  Case  I.,  except  for  the  differ- 
ence in  meaning  of  the  axes  of  reference. 

TABLE  XXXVIII. 


Ability  iu 
—  10 

Average  of 
its  array. 

—  11.6 

Number  of 

cases  in  the 

array. 

2 

Ability  in 
a-!  test. 

+     1 

Average  of 
its  array. 

—     .9 

Number  of 

cases  in  the 

array. 

14 

—    9 

— 

6.3 

3 

+ 

2 

+    3.9 

8 

—    8 

— 

9.5 

2 

_J_ 

3 

+    3.0 

6 

—   7 

— 

1.5 

2 

+ 

4 

+    6.75 

4 

—  6 

—  5 

— 

2.0 
5.55 

2 
9 

+ 
+ 

5 
6 

+  10.2 
+    3.5 

6 
2 

—  4 

—  3 

— 

.6 

8.8 

7 
7 

+ 

+ 

7 
8 

+  13.2 
+    6.3 

6 
3 

o 

— 

2.3 

12 

+ 

9 

+  14.0 

1 

—   1 

+ 

5.5 

4 

+  10 

+    8.0 

3 

0 

— 

.6 

18 

+  14 

+  18.0 

1 

H4 


If  the  measure  of  the  general  tendency  of  each  array  were  deter- 
mined exactly,  the  general  tendency  of  the  relationship  would  be 
exactly  determined  by  the  equations  of  Table  XXXVIII.  and  the 
position  of  the  points  in  Fig.  85.  We  should  have  nothing  to  do 
but  state  them.  For  instance,  suppose  that  in  the  A  a-t  relationship 
the  results  were  : 

A  measure  of  —  5  a-t  words  has  a  related  array  with  a  central  tendency  of  — 10.00 
"  —4  "  "  "  "  "  —   8.00 

"  —3  "  "  "  "  "  —   6.00 

<  <  o  "  "  "  "  "  4  00 

"  i  <<  <<  "  n  "  2.00 


V 


MEASUREMENT  OF  RELATIONSHIPS.  119 


A  measure 

of 

— Oo- 

words  lias  a 

related 

array 

with  a 

cen 

tral  tendency 

of- 

0 

<( 

+  1 

(i 

<< 

it 

(< 

a 

+ 

.50 

(< 

+  2 

a 

" 

" 

it 

(i 

+ 

1.00 

<< 

+  3 

(< 

it 

a 

<< 

it 

+ 

1.50 

a 

+  4 

(< 

" 

n 

" 

a 

+ 

1.50 

it 

+  5 

<< 

it 

a 

a 

it 

+ 

1.50 

These  facts  would  be  the  general  tendency  of  the  relationship. 
One  would  simply  say  : 

"  Individuals  who  in  the  a-t  test  mark  a  given  number  less  than 
the  average  will  in  the  A  test,  on  the  whole,  mark  twice  that  num- 
ber less  than  the  average ;  individuals  at  the  average  in  one  will  be 
at  the  average  in  the  other.  Individuals  marking  1,  2  or  3  more 
than  the  average  in  the  a-t  test  will  mark  one  half  that  number  more 
than  the  average  in  the  A  test.  Individuals  marking  over  +  3  in 
the  a-t  test  do  as  well  and  no  better  in  the  A  test  than  those  mark- 
ing +  3." 

But,  in  fact,  a  relationship  is  almost  never  determined  at  all  ex- 
actly for  each  particular  amount  of  the  first  trait,  especially  not  for 
the  extreme  -f  and  —  amounts.  From  relatively  inexact  measures 
of  the  general  tendencies  of  the  different  arrays  we  infer  the  char- 
acter of  the  relationship  as  a  whole. 

In  making  the  inference  the  first  step  is  to^decide  whether  it  is 
proper  to  assume  that  the  general  tendency  of  the  relationships  is 
uniform  for  all  amounts  of  trait  1,  is  of  the  form  A  =  B  times  a  con- 
stant, is  such  that  the  line  through  the  points  representing  the  exact 
or  true  central  tendencies  of  the  arrays  is  a  straight  line.  In  tech- 
nical terms,  can  it  be  assumed  that  the  correlation  is  rectilinear? 

It  is  clear  that  even  if  the  true  correlation  were  thus  rectilinear, 
the  chance  unreliabilities  of  the  central  tendencies  of  the  actual  ar- 
rays due  to  the  small  number  of  cases  would  make  the  relationship 
vary  in  amount  for  different  arrays.  It  is  also  the  fact  that  mental 
relationships  apparently  approximate  to  the  rectilinear  type  more 
than  to  any  one  other.  It  is,  therefore,  customary  <<>  make  the  as- 
sumption unless  there  is  some  special  reason  for  not  doing  so.  The 
criteria  which  one  mighl  use  to  establish  a  warranted  decision  will 
In'  explained  later.  For  the  present  we  may  best  inquire  what  the 
next  step  in  inference  is,  granted  that  the  true  relationship  is  of  the 
form  A  =B  times  a  constant,  that  the  line  of  correlation  is  rectilinear. 

If  lor  the  true  relationship  A   11=  a  constant,  the  ratio  A   II  will 


120  MENTAL   AND  SOCIAL  MEASUREMENTS. 

be  approximated  by  the  central  tendency  of  all  the  ratios  actually 
found  in  individual  cases  and  the  true  line  of  correlation  will  be  ap- 
proximated closely  by  the  straight  line  from  which  the  points  as  in 
Fig.  85  diverge  least.  In  other  words,  if  we  find  the  central  ten- 
driiiv  of  all  the  individual  ratios  (which  are  given  in  Table  XXXIX.) 
or  the  straight  line  which  fits  best  all  the  points  of  Fig.  85,  we  shall 
have  an  approximation  to  the  true  relationship. 

TABLE  XXXIX. 
Ratios  Expressing  the  Individual  Relationships  of  Table  XXXVII. 


1.90 

5.33 

10.00 

.86 

.40 

4.33 

—  4.00 

1.29 

1.33 

3.67 

—  3.50 

1.43 

.89 

3.33 

-1.00 

1.71 

-   .11 

2.66 

2.00 

2.71 

1.25 

2.00 

3.50 

3.29 

1.13 

—   .67 

3.50 

—  .13 

.43 

7.00 

4.00 

1.25 

00 

6.00 

11.00 

1.25 

1.17 

4.50 

—  3.33 

1.56 

—   .50 

4.00 

—  2.33 

—  .60 

3.60 

4.00 

1.00 

.40 

3.40 

2.00 

2.33 

2.60 

2.60 

2.00 

3.33 

1.29 

2.40 

1.00 

-10.00 

5.00 

1.60 

—  3.50 

—   9.00 

00 

1.00 

—  3.50 

-  8.00 

00 

.60 

—  4.00 

—   7.00 

1.00 

—  1.80 

—  5.50 

—   6.00 

5.75 

—  3.40 

-2.00 

—   5.00 

—  1.20 

3.00 

—  5.00 

—   1.00 

-    .20 

2.25 

—  6.00 

—    1.00 

.80 

2.25 

—  9.00 

2.00 

3.00 

00 

4.00 

4.75 

—   .25 

4.00 

5.20 

—  3.00 

6.00 

.33 

—  3.25 

8.00 

.83 

Each  of  these  methods  has,  however,  a  serious  defect.  In  cal- 
culating the  central  ratio  of  the  observed  individual  ratios,  the  ratios 
for  any  amount  of  one  trait  count  as  much  as  those  for  any  other 
amount.  The  ratio  2.00  obtained  from  a  case  of  —  1  a-t  words 
with  —  2  A's  plays  as  much  of  a  role  as  the  ratio  2.00  from  a  case 
of  —  10  a-t  words  with  —  20  A's.  But  the  latter  should  count 
more,  since  chance  variation  is  far  more  likely  to  deflect  an  individual 
up  or  down  2  .4's  than  20. 


MEASUREMENT  OF  RELATIONSHIPS.  121 

In  calculating  the  straight  line  to  best  fit  the  series  of  points,  a 
point  ascertained  from  an  array  with  few  cases  counts  as  much  as  a 
point  ascertained  from  an  array  with  many.  The  fifth  point  from 
the  right,  due  to  two  cases,  counts  as  much  as  the  sixth  point,  which 
is  due  to  nine  cases.  But,  of  course,  the  knowledge  of  a  relation- 
ship's amount  due  to  nine  cases  is  much  more  reliable  and  deserving 
of  weight  than  that  due  to  two. 

These  difficulties  would  be  removed  if  the  second  method  could  be 
so  modified  that  the  line  drawn  would  be  that  from  which  the  entire 
series  of  points  of  Fig.  84  diverged  least,  or  in  fitter  terms,  would 
be  that  expressing  the  general  tendency  of  relationship  from  which 
all  the  individual  relationships  found  would  most  probably  result. 

The  Pearson  method  of  calculating  the  general  tendency  of  a 
relationship  assumed  to  be  rectilinear  does  this,  and  is,  therefore,  a 
method  of  the  utmost  service  to  the  student  of  causal  and  other 
relationships  in  the  mental  sciences.  The  formula  used  and  con- 
venient ways  of  making  the  necessary  calculations  will  be  explained 
later. 

The  final  desideratum  in  the  measurement  of  a  relationship  is 
that  it  be  intelligible  in  itself  and  commensurable  with  measure- 
ments of  other  relationships. 

It  is  obviously  misleading  to  say  that  a  girl  who  is  14  above  in 
the  a-t  test  and  26  above  in  the  A  test  is  186  per  cent,  as  far  above 
in  the  latter  as  in  the  former.  In  both  cases  she  is  the  best  girl  of 
the  group  and  is  in  reality,  therefore,  equally  far  above  the  average. 
Similarly,  girls  who  were  -f-  4  in  the  a-t  and  +  9  in  the  A  test  would 
really  be  equally  superior  in  both,  for  they  would  be  in  both  the  23d 
to  26th  persons  from  the  top  out  of  the  122.  Distance  from  the 
average  in  each  case  must,  if  the  two  cases  are  to  be  commensurate, 
be  in  terms  of  the  variability  of  the  distribution.  The  variabilities 
are  :  a-t  test,  A.  D.  =  3.57  ;  A  test,  A.  D.  =  8.33.  Case  1  on  one 
list  should  really  be  scored  —  10./3.57  and  —  19/8.33,  giving  the 
ratio  .82.     The  table  of  ratios  thus  corrected  becomes  Table  X  L. 

The  diagram  may  be  corrected  by  dividing  each  measure  by  the 
variability  of  the  distribution  to  which  it  belongs,  or  more  easily  by 
arranging  the  scale  on  the  diagram  so  as  to  make  the  proper  allow- 
ance and  then  using  the  original  figures. 

The  difficulty  in   comparing  different  relationships,  due  to  the 


122  MENTAL  AND  SOCIAL  MEASUREMENTS. 


TABLE  XL. 

jj  Ratios 

of  Table  XXXIX.  Corrected  for 

THE    VAU 

of  each  Trait. 

82 

229 

429 

37 

17 

186 

—172 

55 

57 

157 

—150 

61 

38 

143 

—  43 

73 

-    5 

114 

86 

116 

54 

86 

150 

141 

48 

—  29 

150 

—    6 

18 

300 

172 

54 

00 

257 

472 

54 

50 

193 

—143 

67 

-  21 

172 

—100 

—  26 

154 

172 

43 

17 

146 

86 

100 

112 

112 

86 

143 

55 

103 

43 

—429 

215 

69 

—150 

-386 

00 

43 

—150 

—343 

00 

26 

—172 

—300 

43 

-  77 

—236 

—257 

247 

-146 

—  86 

—215 

—  51 

129 

—215 

—  43 

—    9 

97 

—257 

—  43 

34 

97 

—386 

86 

129 

00 

172 

204 

-  11 

172 

223 

129 

257 

14 

139 

343 

36 

fact  that  the  units  of  measure  for  the  different  traits  are  incommensu- 
rate, disappears  if  they  are  each  and  all  reduced  to  terms  of  the 
variability  of  the  group.  They  then  become  commensurate,  indeed 
identical,  in  the  sense  that  in  each  of  the  tests  the  best  person  of 
10,000  chosen  at  random  would  be  plus  the  same  figure.  The  10th 
best  in  the  one  would  be  plus  the  same  amount  as  the  10th  best  in 
any  other,*  etc. 

The  estimation  of  any  relationship  for  a  group  would  then  be 
comparable  with  that  of  any  other  relationship  for  that  group,  and 
many  now  awkward  questions  of  the  mental  and  social  sciences  would 
be  amenable  to  exact  and  readily  obtained  answers.  The  Pearson 
method  of  calculating  rectilinear  relationships  fulfills  this  desider- 
atum,  and  thus  meets   the   exacting  demands  of  a   measure  of  the 

*This  would  hold  exactly  only  in  so  far  as  the  forms  of  distribution  of  the  dif- 
ferent traits  were  alike. 


MEASUREMENT  OF  RELATIONSHIPS.  123 

general  tendency  of  a  relationship  between  two  variable  quantities 
with  unknown  zero  points  and  units  directly  incommensurable. 

The  Pearson  method  obtains  as  its  measure  of  the  relationship  a 
single  number,  which  may  be  anywhere  between  1.00  and  —  1.00. 
A  coefficient  of  correlation  between  two  abilities  of  +  100  per  cent, 
means  that  the  individual  who  is  the  best  in  the  group  in  one  ability 
will  be  the  best  in  the  other,  that  the  worst  man  in  the  one  will  be 
the  worst  in  the  other ;  that  if  the  individuals  were  ranged  in  order 
of  excellence  in  the  first  ability  and  then  in  order  of  excellence  in  the 
second,  the  two  rankings  would  be  identical ;  that  any  one's  station 
in  the  one  will  be  identical  with  his  station  in  the  other  (both  ^being 
reduced  to  terms  of  the  variabilities  of  the  abilities  as  units  to  allow 
comparison).  A  coefficient  of  —  100  per  cent,  would,  per  contra, 
mean  that  the  best  person  in  the  one  ability  would  be  the  worst  in 
the  other,  that  any  degree  of  superiority  in  the  one  would  go  with  an 
equal  degree  of  inferiority  in  the  other,  and  vice  versa.  A  coefficient 
of  +  62  per  cent,  would  mean  that  (comparison  being  rendered  fair 
here  as  always  by  reduction  to  the  variabilities  as  units)  any  given 
station  in  the  one  trait  would  imply  62  hundredths  of  that  station  in 
the  other.  A  coefficient  of  —  62  would,  of  course,  mean  that  any 
degree  of  superiority  would  involve  62  hundredths  as  much  inferior- 
ity, and  vice  versa. 

The  method  of  calculating  the  Pearson  coefficient  of  correlation 
is  to  multiply  each  case's  deviation  from  the  average  in  the  one  trait 
by  its  deviation  from  the  average  in  the  other  trait ;  to  add  together 
all  the  products  thus  found  and  divide  their  sum  by  the  number  of 
cases  times  the  standard  deviation  of  the  first  trait  times  the  standard 
deviation  of  the  second  trait.     That  is,  the  coefficient  of  correlation, 

r = ^y 

naxa2' 

The  arithmetic  involved  in   calculating  Pearson    coefficients  is 

simple,  and,  though  lengthy,  does  not  take  so  long  a  time  as  might  be 

supposed.     The  apparently  tedious  process  of  multiplication  can  be 

done    quickly    and    with    no   mental   effort  by  the  use  of  Crelle's 

Rechentafeln,*    which   is   a   multiplication  table   running  to   1,000 

times  1,000. 

*  Published  by  Georg  Reimer,  Berlin.  The  price  is  about  $4.50.  A  multipli- 
cation table  up  to  100  times  1 0(1  is  given  in  Appendix  I.  of  this  book. 


124 


MENTAL   AND  SOCIAL  MEASUREMENTS. 


The  squaring  involved  in  the  calculation  of  <rx  and  <r„  is,  of  course, 
done  with  the  aid  of  a  table  of  squares,  such  as  Barlow's  tables.* 
The  addition  is  tedious  unless  one  has  at  his  service  an  adding 
machine.  Even  without  an  adding  machine,  however,  a  coefficient 
can  be  calculated  from  1,000  individual  relationships  under  the  most 
unfavorable  circumstances  in  less  than  a  day.  Different  ways  of 
arranging  the  material  economize  time  in  different  cases.  The  pro- 
cedure which  is  most  generally  serviceable  is  to  calculate  the  average 
for  each  array  and  then  replace  —xy  by  [(av.  of  first  array  of  B)  x 
(amount  of  A  with  which  it  is  correlated)  x  (its  number  of  cases)]  + 
[(av.  of  second  array  of  B)  x  (amount  of  A  with  which  it  is  corre- 
lated) x  (its  number  of  cases)]  etc.,  through  the  last  array.  This 
reduces  the  addition  in  part  to  multiplication  and  gives  us  knowledge 
of  the  degree  to  which  the  relationship  approaches  a  rectilinear  form. 
Thus  in  the  case  of  our  illustration  we  obtain  the  facts  of  Table  XLI. 


TABLE 

XLI. 

A 

B 

c 

Various  amounts 
of  Trait  1. 
a-l  words. 

Averages  of 
related  arrays. 

Number  of 

cases  in 
the  arrays. 

„— 

10 

— 

11.50 

2 

— 

9 

— 

6.33 

3 

— 

8 

— 

9.50 

2 

— 

7 

— 

1.50 

2 

— 

6 

— 

2.00 

2 

— 

5 

— 

5.56 

9 

— 

4 

— 

.57 

7 

— 

3 

— 

8.86 

7 

— 

2 

— 

2.33 

12 

— 

1 
0 

i 

5.50 

4 

18 

+ 

1 

— 

.93 

14 

+ 

2 

+ 

3.88 

8 

+ 

3 

+ 

3.00 

6 

+ 

4 

+ 

6.75 

4 

+ 

5 

+  10.17 

6 

+ 

6 

+ 

3.50 

2 

_L 

7 

+  13.17 

6 

+ 

8 

+ 

6.33 

3 

+ 

9 

+  14.00 

1 

+  10 

+ 

8.00 

3 

+  14 

+  18.00 

1 

D 

Averages  of  array 

times  amount  of 

Trait  1  times  frequency, 

i,  e.,AX-BX& 

230 

171 

152 

21 

24 

250 

16 

186 

56 

22 

0 

13 

62 

54 

108 

305 

42 

553 

152 

126 

240 

252 

^965 

*The  squares  and  square  roots  of  the  numbers  up  to  1,000  are  given  in  Appen- 
dix IT.  of  this  book. 


MEASUREMENT  OF  RELATIONSHIPS.  125 

The  relationship  may  fairly  be  assumed  to  be  rectilinear  from  the 
figures  in  column  B,  and  their  graphic  representation  in  Fig.  85. 
Using  the  Pearson  formula,  then,  we  have 

Sxy  =  2,965.     <rl  =  4.65  (see  calculation  on  page  126). 

<72=10.1         "  "  "        " 

n=  122 

r,  the  coefficient  of  correlation,  then  equals  +  .52.  If  for  other 
reasons  it  is  known  to  be  valid  to  assume  rectilinear  correlation,  it  is 
somewhat  quicker  to  calculate  -xy  directly  from  the  individual  rec- 
ords. This  calculation  in  full,  together  with  that  of  ox  and  a2  is  given 
in  Table  XLII.  Ordinary  arithmetical  skill  could  much  abbreviate 
the  calculation  given  there  by  combining  multiplicands  in  multiply- 
ing and  so  saving  later  addition. 


TABLE 

XLII. 

A.— 

Calculation  of  2xy. 

190 

48 

0 

10 

42 

40 
108 

39 
33 

0 
0 

—  16 

—  14 

63 

70 

72 

30 

0 

—  4 

84 

—  9 

24 

0 

8 

133 

80 

18 

0 

14 

161 

72 

—  6 

0 

14 

21 

28 

0 

16 

80 

0 

24 

0 

44 

80 

42 

-18 

18 
16 

0 
0 

—  30 

—  21 

126 

90 

16 

0 

9 

40 

85 

8 

0 

21 

260 

65 

8 

0 

30 

252 

60 

4 

-10 

45 

40 

—  14 

-  9 

0 

25 

—  14 

-  8 

0 

15 

—  45 

—  85 

—  16 

—  22 
2 

-  7 

-  6 

-  5 

16 
92 

—  30 

48 

—  5 

-  1 

—  5 

36 

—  6 

-  1 

30 

36 

—  9 

2 

75 

0 

—  4 

—  48 

—  52 

0 
0 
0 

4 
4 
6 
8 

115 

130 

12 

30 

—  60 


The  sum  of  the  xy  products  =  2,965. 


126  M i:\TAL  AXD  SOCIAL  MEASUREMENTS. 

B.  —  Calculation  of  <Tj  and  a2. 
—  202  x  1    =  400  — 102  X    2  =  200 


— 11)2 

1 

381 

— 18* 

2 

648 

—  IT2 

1 

289 

—  152 

1 

225 

—  142 

2 

392 

—  132 

5 

845 

_12« 

1 

144 

-   112 

5 

605 

—  102 

6 

600 

—     92 

7 

567 

—   82 

5 

320 

—  72 

4 

196 

—   62 

3 

108 

—  52 

4 

100 

—  42 

3 

48 

—  32 

2 

18 

22 

4 

16 

—     I2 

4 

4 

0 

5 

0 

+    I2 

5 

5 

+     22 

3 

12 

+  32 

6 

54 

+    42 

3 

48 

+    53 

4 

100 

+    62 

6 

216 

+    72 

3 

147 

+    82 

4 

256 

+   92 

5 

225 

+  102 

1 

100 

+  112 

2 

242 

+  132 

1 

169 

-142 

3 

588 

+  162 

1 

256 

+  172 

1 

289 

+  182 

1 

324 

+  202 

1 

400 

+  21* 

1 

441 

—  222 

3 

1,452 

-  25-' 

2 

1,250 

12,480 

52,480^-122  =  102.3 

^102.3  =  10.1 

iVV2  =  122  X  10.1  X  4.65 

JVffjffj  =  5730 

r  =  2965/5730,  r  =  +  .52 

—  92 

3 

243 

-   82 

2 

128 

—   72 

2 

98 

—   6* 

2 

72 

—   52 

9 

225 

—  42 

7 

112 

—   3* 

7 

63 

22 

12 

48 

—   I2 

4 

4 

0 

18 

0 

+    l2 

14 

14 

+    22 

8 

32 

+    32 

6 

54 

+    43 

4 

64 

+    52 

6 

150 

+    62 

2 

72 

+    72 

6 

294 

+     82 

3 

192 

+  92 

1 

81 

+  102 

3 

300 

+  142 

1 

196 

2,642 

2,642  -=-122  =  21.656 

^21.656  =  4.65 

MEASUREMENT  OF  RELATIONSHIPS.  127 

All  the  discussion  of  measurements  of  relationship  so  far  presup- 
poses that  the  facts  related  are  measured  exactly.  There  will,  how- 
ever, in  mental  and  social  measurements  commonly  be  a  considerable 
error  in  each  individual  fact  of  those  to  be  related.  For  instance, 
in  our  illustration  the  '  A's  marked  by  each  individual '  is  a  score  de- 
pending upon  only  one  trial  of  60  seconds.  With  many  trials  on 
many  different  occasions,  the  individuals  concerned  would  attain 
somewhat  different  measures.  So  also  with  the  '  a-t  words  marked.' 
Let  us  call  raiOC  m  the  r  which  would  be  obtained  in  our  illustration 
from  accurate  measures  in  both  traits  for  all  of  the  individuals,  and 
rapp  m  the  r  which  is  in  fact  calculated  from  the  single  measures. 
racc.m.  wiH  be  greater  *  than  rapp  m ,  for  the  influence  of  chance  inac- 
curacy in  the  measures  to  be  related  is  always  to  produce  zero  cor- 
relation. If  two  series  of  pairs  of  values  are  due  entirely  to  chance 
the  correlation  will  be  zero,  and  in  so  far  as  they  are  at  all  due  to 
chance,  they  will  reduce  the  correlation. 

The  chance  variation,  which  in  the  long  run  cuts  its  own  throat 
in  the  case  of  averages  and  variabilities,  can  not  in  the  case  of  a 
relationship  be  thus  rendered  innocuous  by  mere  numbers.  For 
instance  the  true  relationship  between  the  volume  of  bodies  of  water 
at  constant  pressure  and  temperature,  etc.,  and  their  weight  is  +  1.00. 
Suppose  now  that  the  true  measures  for  ten  pairs  were  : 


Case. 

Vol. 

wt. 

A 

2 

4 

B 

4 

8 

0 

6 

12 

D 

7 

14 

E 

8 

10 

F 

9 

18 

G 

10 

20 

H 

11 

22 

I 

13 

20 

J 

15 

30 

The  correlation  is  evidently  +  1 .00. 

Suppose  the  person  measuring  them  got  instead  of  these  figures 
certain  chance  variations  from  them  due  to  the  error  of  his  measuring. 

If  the  reader  will  distribute  by  chance  among  these  20  errors, 
say  5  of  1,  5  of  —  1,  4  of  2,  4  of  —  2,  1  of  3  and  1  of  —  .">  and  then 

*  By  greater  is  meant  more  plus  if  the  relationship  from  accurate  measures  is 
positive,  more  minus  if  it  is  negative. 


L28  MENTAL    AND  social  MEASUREMENTS. 

calculate  again  the  coefficient,  he  will  find  it  to  be  loss  than  before. 
It'  he  will  let  the  chance  errors  be  larger,  e.g.,  5  each  of  -f  2  and 
—  2,  1  each  of  +  -1  and 1  and  1  each  of  +  6  and  —  6,  the  coeffi- 
cient will  be  still  more  reduced.  The  same  will  hold  regardless  of 
whether  1"  or  10,000  pairs  of  related  values  are  taken. 

To  correct  for  this  '  attenuation '  of  the  coefficient  by  chance  errors 
in  the  data,  it  is  necessary  to  have  at  least  two  independent  measures 
o(  the  measures  to  be  related.  When  these  are  at  hand  the  pro- 
cedure is  as  follows: 

Let  A  and  B  be  the  traits  to  be  related. 

Let  p  be  a  series  of  exact  measures  of  A. 

Let  q  be  the  related  series  of  exact  measures  of  B. 

Denote  by  r     the  coefficient  of  correlation  of  A  and  B,  obtain- 

J        pq 

able  from  the  two  series  p  and  q.  rpq  is  thus  the  required  real  rela- 
tionship. 

Denote  bv  r  ,  ,  the  average  of  the  correlations  between  each  series 
of  values  obtained  for  trait  A  and  each  series  of  related  values  for 
trait  B. 

Denote  by  r  ,  „  the  average  of  the  correlations  between  any  one 
series  of  measures  of  trait  A  and  any  other  corresponding  series  of 
independent  measures  of  trait  A. 

Denote  by  r  ,  „  the  average  of  the  correlations  between  any  one 
series  of  measures  of  trait  B  and  any  other  corresponding  series  of 
independent  measures  of  trait  B. 

Then/-     = 


V   pp    A    qq    / 

Thus  if  we  have  two  series  of  independent  measures  of  trait  A 
and  similarly  of  the  related  trait  B,  if,  that  is,  we  have  certain  indi- 
viduals measured  twice  in  each  trait,   we  shall  have  as  our  formula 

4 

in  which  pl  and  p„  refer  to  the  two  independent  series  of  measures  of 
trait  A  ;  qx  and  q2  refer  to  the  two  independent  series  of  measures  of 
trait  B  ;  r„  „  is  the  coefficient  of  correlation  between  the  first  and 
second  measures  of  A  ;  r       is  the  coefficient  of  correlation  between 


MEASUREMENT  OF  RELATIONSHIPS.  129 

the  first  and  second  measures  of  B ;  rpm  is  the  coefficient  of  correla- 
tion between  the  first  measure  of  A  and  the  first  measure  of  B ;  /y,i7j 
is  the  coefficient  of  correlation  between  the  first  measure  of  A  and 
the  second  measure  of  B  ;  r;),,f/i  is  the  coefficient  of  correlation  between 
the  second  measure  of  A  and  the  first  measure  of  B  ;  rpm  is  the  coef- 
ficient of  correlation  between  the  second  measure  of  A  and  the  second 
measure  of  B. 

A  second  method  *  of  allowing  for  the  inaccuracy  of  the  original 
measures  of  the  facts  to  be  related  is  based  upon  the  obvious  fact 
that  an  increase  in  the  number  of  measures  of  each  of  such  facts  in- 
creases its  accuracy.  From  the  increase  in  the  closeness  of  the 
relationship  as  we  use  the  central  tendency  of  2,  3,  4,  5  .  .  .  trials 
of  each  individual,  we  may  prophesy  what  the  relationship  would  be 
if  we  had  at  hand  measures  from  so  many  trials  of  all  the  individuals 
as  to  give  the  central  tendencies  exactly. 

Let  r  be  the  coefficient  of  correlation  that  would  be  found  if  the 
measures  of  the  related  facts,  A  and  B,  were  perfectly  exact. 

Let  m  be  the  number  of  independent  measures  of  A,  p{p^p3,  etc. 

Let  n   «     «  «        "  "  "  "  B,  qxq./y  etc. 

Let  r  , ,  be  the  average  of  the  correlations  between  each  series  of 
values  obtained  for  trait  A,  with  each  series  obtained  for  trait  B. 

Let  r  „  ,,  be  the  correlation  obtained  when  pxp2pv  etc.  are  com- 
bined to  give  the  measure  of  trait  A,  when,  that  is,  each  individual 
is  represented  by  his  most  likely  central  tendency  in  trait  A,  and 
when  qxq2q3  are  similarly  combined  to  give  the  measure  of  trait  B. 

Then  r    =  '"'        -±1 

i"l  4/  t 

v  mn  —  I 

Useful  as  these  formula?  for  correction  of  attenuation  due  to  inac- 
curate measures  are,  it  is  wise  not  to  overwork  them  by  substituting 
their  use  for  the  attainment  of  reasonably  precise  original  measures. 
The  beginner,  at  all  events,  may  best  work  here  only  with  original 
measures,  the  P.  E.true_obtalnedt  of  which  is  not  over  5  per  cent,  of 
their  amount. 

Another  source  of  error,  a  much  less  important  one  in  practice, 

*For  a  farther  description  of  this  method  and  the  first  method  as  well  sic  the 
article  in  the  Am.  J.  of  Pmj.,  for  January,  1904,  by  C.  Spearman,  to  whom  the 
formulae  are  due. 

tSee  next  chapter  for  the  explanation  of  this  term. 

9 


130  MENTAL   AND  SOCIAL  MEASUREMENTS. 

is  the  inaccuracy  of  the  central  tendencies  from  which  the  deviations 
ani  measured.  The  true  relationship  is  of  course  that  existing 
between  the  deviations  of  one  series  of  measures  from  their  true 
central  tendency  and  the  corresponding  deviations  of  the  second 
series  from  their  true  central  tendency.  The  effect  of  inexact 
measures  of  the  central  tendencies  is  to  make  the  obtained  coefficient 
larger  *  than  the  true  coefficient  when  the  inaccuracies  are  both  in  the 
same  direction  and  smaller  when  they  are  in  different  directions. 
The  error  is  "inconsiderable  for  inaccuracies  such  as  occur  in  central 
tendencies  calculated  from  100  or  more  individual  measures. 

A  third  source  of  error  deserves  mention,  though  it  is  logical 
rather  than  statistical.  To  measure  the  relation  between  quality  A 
and  quality  B,  we  should  have  a  series  of  pairs  of  amounts  related  only 
through  the  relationship  of  A  to  B.  But  unless  great  care  is  taken 
in  the  selection  of  the  data,  other  factors  affecting  the  relationship  of 
the  amounts  are  sure  to  enter.  Thus  in  relating  mental  capacities, 
if  we  use  children  of  different  ages,  the  factor  of  age,  as  well  as  the 
intrinsic  relationship  between  the  traits,  is  at  work.  The  real  rela- 
tion between  a  city's  lighting  and  its  need  of  police  protection  might 
be  inverse  but  actual  correlations  of  the  per  capita  expense  for  the 
two  items  in  American  cities  might  show  a  direct  relationship  due  to 
the  entrance  of  the  factor,  municipal  expensiveness  as  a  whole.  The 
influence  of  heredity  can  not  be  inferred  from  fraternal  correlation 
until  a  discount  is  made  for  the  factor,  similar  training.  Means  of 
correcting  for  irrelevant  factors  have  been  devised,  but  it  is  safest  to 
get  data  free  from  them  in  the  first  instance. 

On  page  119  the  problem/  How  to  decide  whether  a  relationship 
may  be  assumed  to  be  rectilinear  ? '  was  suggested  and  postponed. 
It  can  not  be  given  an  absolute  answer.  One  can,  by  knowing  the 
unreliability  of  each  array's  central  tendency,  measure  the  likelihood 
that  any  given  straight  line  chosen  could  be  the  true  line  of  correla- 
tion. But  some  slightly  crooked  line  would  have  a  still  greater  like- 
lihood. So  far  as  the  figures  go,  the  most  likely  true  relationship  is 
the  crooked  line  that  passes  through  every  point.  It  is  because  of  a 
general  confidence  that  nature  is  simple  rather  than  complex,  that 
regularity  in  relationships  is  more  likely  than  irregularity,  that  we 

*  By  larger  is  meant  more  plus  in  case  the  coefficient  is  positive,  more  minus  in 
case  it  is  negative  ;  by  smaller  is  meant  the  reverse. 


MEASUREMENT  OF  RELATIONSHIPS.  131 

assume  that  the  unevenness  of  the  correlation  found  would  disappear 
with  more  cases.  If  the  student  plots  the  line  of  central  tendencies 
of  arrays  and  on  either  side  of  it  a  line  at  the  distance  from  it  of  the 

**'     ^*tnie  central  tendency -obtained  central  tendency?         aUCl      tUen      UnaS     tiiat      tile 

straight  line  which  best  fits  the  central  tendency  points  falls  in  nine 
out  of  ten  cases  within  the  P.  E.  lines,  he  will  rarely  be  wrong  in 
assuming  correlation  to  be  rectilinear. 

If  correlation  is  demonstrably  not  rectilinear  the  mode  of  express- 
ing its  nature  and  amount  will,  of  course,  vary.  The  general  prob- 
lem will  be,  as  always,  to  express  the  general  tendency  of  relation- 
ship from  which  the  actually  found  relationships  can  be  derived  with 
least  improbability.  Acquaintance  with  the  concrete  data  concerned 
and  natural  ingenuity  and  insight  will  here  be  of  far  more  service 
than  cut  and  dried  methods  of  technical  procedure. 

In  presenting  results  no  Pearson  coefficient  or  other  single  ex- 
pression should  be  given  without  also  the  total  correlation  table,  or 
at  least  a  diagram  or  list  of  the  averages  of  the  arrays  such  as  may 
enable  the  reader  to  judge  how  far  the  relationship  throughout  is 
that  expressed  by  the  single  ratio. 

The  facts  to  be  related  in  the  mental  and  social  sciences  may  be 
either  (1)  the  varying  conditions  of  a  trait  in  an  individual  (to  be  re- 
lated to  corresponding  conditions  in  him  of  some  other  trait)  or  (2) 
the  varying  conditions  of  a  trait  found  in  different  individuals  of  a 
group  (to  be  related  to  the  conditions  found  in  some  other  trait  in 
the  same  individuals)  or  (3)  the  varying  central  tendencies  of  a  trait 
found  in  different  subgroups  of  a  larger  group  or  collection  of  groups 
(to  be  related  to  the  central  tendencies  found  in  the  case  of  some  other 
trait  in  the  same  subgroups). 

For  example,  one  may  seek  (Case  1)  the  relation  between  the 
quickness  of  perception  of  an  individual  at  various  times  and  his 
quickness  of  movement  at  corresponding  times.  Or  one  may  seek 
(Case  2)  the  relation  between  the  quickness  of  perception  in  general 
of  Jones,  Smith,  Brown,  etc.,  and  the  quickness  of  movement  pos- 
sessed in  general  by  the  same  individuals.  Or  (Case  3)  one  may 
seek  the  relationship  between  the  general  quickness  in  perception  of 
races  to  their  quickness  of  movement. 

*The  meaning  of  this  quantity  may  he  left  undefined  until  the  next  chapter  is 
read. 


L32  MENTAL  AND  SOCIAL   MEASUREMENTS. 

It  should  be  noted  that  the  difference  in  the  three  cases  is  not  in 
the  mere  number  of  individuals  studied.  The  essential  difference 
would  remain  if  we  used  a  million  cases  to  determine  the  relationship 
of  two  traits  within  an  individual,  only  a  hundred  thousand  to  de- 
termine the  relationship  among  individuals  and  only  ten  thousand  to 
determine  it  for  races.  The  essential  difference  is  in  the  questions 
to  be  solved.  From  them  it  follows  also  that  in  Case  1  if  several  in- 
dividuals are  studied  a  number  of  pairs  of  figures  for  each  individual 
will  be  used  and  the  general  tendency  of  the  relationship  in  each  in- 
dividual will  be  worked  out  separately.  If  the  results  from  different 
individuals  are  then  combined  they  will  be  combined  as  a  group  of 
facts  according  to  the  methods  of  Chapter  IV.  In  Case  2,  on  the 
contrary,  a  single  pair  of  figures  will  represent  the  relationship  in 
anyone  individual  and  these  pairs  will  be  combined  according  to  the 
method  of  the  present  chapter.  In  Case  3  a  single  pair  of  figures 
will  represent  the  relationship  in  each  subgroup. 

The  problem  of  measurement  itself  is  the  same  for  three  cases, 
the  difference  being  in  the  data  used  and  the  consequent  meaning  of 
the  coefficient  of  correlation  obtained.  To  any  one  of  the  following 
series  of  related  pairs  the  mode  of  procedure  discussed  in  this  chap- 
ter is  applicable. 

Related  by  Identity  of  Conditions. 

Trait  T  and  trait  TY  in  individual  .1  under  conditions  Cv 

"        «         "  C2 

<<      «  c3 

Related  by  Identity  of  the  Individual. 

Trait  T  and  trait  Tx  in  group,  ten-year-olds,  in  individual  Jx 

a  a  n  t 

±2 

a  a  a  T 

Related  by  Identity  of  the  Subgroup. 
Trait  T  and  trait  2\  in  group,  all  men,  in  subgroup  Chinese. 
"         "         "  Negroes. 

"         "         "  Indians. 

It  is  perhaps  needless  to  point  out  that  the  existence  of  a  certain 
relationship  within  an  individual  does  not  imply  anything  about  the 
relationship  within  a  group  of  individuals,  nor  that  again  about  the 
relationship  within  a  group  of  groups.  Individuals  may  be  happier 
when  they  are  richer,  but  rich  individuals  amongst  Americans  may 
be  no  happier  than  poor  individuals,  and  from  neither  fact  could  we 


MEASUREMENT  OF  RELATIONSHIPS.  133 

infer  that  the  American  population  would  be  happier  or  less  happy 
than  the  Chinese  or  the  Negro  population. 

For  similar  reasons  the  nature  and  amount  of  a  relationship  will 
depend  upon  the  group  selected.  If,  for  instance,  the  relationship 
between  knowledge  of  history  and  knowledge  of  English  literature 
is  measured  in  the  group,  high-school  graduates,  by  using  the 
deviations  of  individuals  from  the  high-school  graduates'  averages  in 
the  two  traits,  the  relationship  will  be  less  close  than  if  we  use  the 
group,  all  people.  The  relationship  between  height  and  weight  will 
be  less  close  if  measured  in  the  group,  18-year-olds,  than  if  measured 
in  all  children  under  twenty.  Any  relationship  so  calculated  should 
always  be  thought  of  as  the  relationship  of  deviations  from  the 
averages  in  the  two  traits  in  the  individuals  of  the  group  in  question. 
To  assume  that  the  relationship  found  in  any  given  group  holds 
good  also  for  a  different  group  is  valid  only  if  the  given  group  is  a 
random  selection  from  the  other  group. 

Application  of  the  Theory  of  Measurements  of  Variable  Relationships 
to  the  Problem  of  Measuring  Mental  Inheritance. 

The  measurement  of  mental  inheritance  involves  the  measure- 
ment of  similarities  between  related  individuals  and  the  measurement 
of  the  amount  of  such  similarity  to  be  attributed  to  training.  The 
first  problem  is  statistically  identical  with  that  of  measuring  the  re- 
lationship between  two  mental  traits,  only  here  the  two  traits  will  be 
the  same  trait  in  two  related  individuals,  and  the  coefficient  of  cor- 
relation will  measure  not  the  implication  of  one  trait  with  respect  to 
another  in  the  same  man,  but  the  implication  of  one  trait  in  one  man 
with  respect  to  the  same  trait  in  his  relative.  In  the  formula,  that 
is,  the  xy  products  will  be  each  the  product  of  one  person's  deviation 
and  that  of  his  relative  ;  nx  will  be  the  variability  of  all  the  first 
members  of  the  series  of  related  pairs  and  a2  the  variability  of  all 
the  second  members.     N  will  be  the  number  of  pairs. 

Application  to  the  Study  of  ('mi sal  Relationships. 

The  possibility  of  measuring  relationships  conveniently  and  pre- 
cisely is  one  step  toward  the  study  of  causes  in  the  mental  sciences. 
It  gives  us  a  means  of  making  Mill's  method  of  'concomitant  varia- 
tions' exact  and  applicable  to  variable  facts.      It  allows  us  to  make 


134  MENTAL  AND  SOCIAL    MEASUREMENTS. 

use  of  the  criterion  that  the  cause  must  be  equal  to  the  effect. 
Whenever  oue  finds  two  quantities  correlated  he  may  properly  pro- 
ceed to  test  the  hypotheses  that  one  causes  the  other  in  part  and  that 
both  are  due  in  part  to  some  common  cause. 

The  point  of  view  of  this  long  chapter  may  be  summed  up  in  a 
few  short  practical  precepts.     They  are  : 

Think  what  you  are  relating,  and  that  any  relationship  is 
measured  by  a  series  of  ratios. 

If  the  measures  are  absolute  amounts,  bear  in  mind  the  signifi- 
cance of  the  zero  points  from  which  they  are  measured. 

If  the  measures  are  deviations  from  some  central  tendency,  bear 
in  mind  the  nature  of  the  group  whose  central  tendency  it  is. 

Keep  before  you  always  the  total  series  of  ratios  found. 

Do  not  be  satisfied  with  crude  means  of  measuring  any  presum- 
ably rectilinear  relationship.  The  Pearson  coefficient  requires  not 
much  more  time  and  is,  for  both  exactness  and  convenience,  far 
superior. 

Problems. 

29.  Calculate  the  relationship  between  changes  in  pauperism  and 
changes  in  out-relief  from  the  following  data  :  * 

Percentage  Ratios  of  Pauperism. 

105-115    115-125 


15-25 

25-35 

35-45 

45-55 

55-65 

65-75 

75-S5 

85-95 

95-K 

15-25 

1 

o     25-35      1 

1 

4 

1 

"•§     35-45 

3 

2 

10 

3 

3 

rt    45-55 

2 

4 

7 

8 

6 

4 

£    55-65 

4 

10 

11 

11 

8 

§    65-75 

4 

10 

13 

7 

2 

1 

*>    75-85 

1 

7 

12 

8 

1 

7 

1 

O    85-95 

1 

4 

3 

1 

1 

o    95-105 

1 

4- 

5 

4 

5 

S  105-115 

1 

4 

5 

1 

'■§  115-125 
^  125-135 

1 
1 

3 

1 
1 

3 
1 

1 
1 

§>  135-145 

1 

g  145-155 

g  155-165 

£  165-175 

175-185 

185-195 

1 

*From  an  article  by  G.  Udny  Yule,  in  the   Journal  of  the   Royal   Statistical 
Society,  Vol.  62,  p.  281. 


MEASUREMENT  OF  RELATIONSHIPS.  135 

Each  figure  iu  the  table  represents  the  number  of  cases  of  the  re- 
lationship denoted  by  the  figure  above  it  in  the  horizontal  scale  taken 
with  the  figure  opposite  it  in  the  vertical  scale.  Thus  the  second 
column  reads :  '  Of  districts  having  a  change  of  25-35  in  pauper- 
ism, one  had  a  change  of  25-35  in  out-relief  ratio,  three  had 
changes  of  35-45  in  out-relief  ratio,  and  2  had  changes  of  45-55. 


CHAPTER  X. 

THE    RELIABILITY    OF    MEASURES. 

When  from  a  limited  number  of  measurements  of  an  individual 
fact,  say  of  A's  monthly  expenses  or  B's  ability  in  perception,  we 
calculate  its  average,  the  result  is  not,  except  by  chance,  the  true 
average.  For,  obviously,  one  more  measurement  will,  unless  it 
happens  to  coincide  with  the  average  obtained,  change  it.  For  in- 
stance, the  first  30  measures  of  iJ's  ability  in  reaction  time  gave 
the  average  .1405;  the  next  seven  measures  being  taken  into  ac- 
count, the  average  became  .1400  ;  with  the  next  seven  it  became 
.1406  —  ;  with  the  next  seven,  .1406  -f.  By  the  true  average  we 
mean  the  average  that  would  come  from  all  the  possible  tests  of  the 
trait  in  question.  The  actual  average  obtained  from  a  limited  finite 
number  of  these  measures  is,  except  by  chance,  only  an  approxi- 
mation toward  the  true  average.  So  also  with  the  accuracy  of  the 
measure  of  variability  obtained.  The  true  variability  is  that  mani- 
fested in  the  entire  series  of  measurements  of  the  trait ;  the  actually 
obtained  variability  is  an  approximation  toward  it.  The  true  aver- 
age and  the  true  variability  of  a  group  mean  similarly  the  measures 
obtained  from  a  study  of  all  the  members  of  the  group. 

It  is  necessary,  then,  to  know  how  many  trials  of  an  individual, 
how  many  members  of  a  group,  must  be  measured,  to  obtain  as  ac- 
curate knowledge  as  we  need.  Or,  to  speak  more  properly,  it  is 
necessary  to  know  how  close  to  the  true  measure  the  result  obtained 
from  a  certain  finite  number  of  measures  will  be. 

It  is  clear  that  the  true  average  of  any  set  of  measures  is  the 
average  calculated  from  all  of  them.  If  the  average  we  actually 
obtain  is  calculated  from  samples  chosen  at  random,  it  will  probably 
diverge  somewhat  from  the  average  calculated  from  all.  So  also 
with  obtained  and  true  measures  of  total  distribution,  variability,  of 
difference  and  of  relationship.  We  measure  the  unreliability  of  any 
obtained  measure  by  its  probable  divergence  from  the  true  measure. 

It  is  clear  also  that  the  divergence  of  any  measure  due  to  a 
limited  number  of  measures  from  the  corresponding  measure  due  to 
the  entire  series,  will  vary  according  to  what  particular  samples  we 

136 


THE  RELIABILITY  OF  MEASURES.  137 

hit  upon,  and  that  if  the  samples  are  taken  at  random  this  variation 
in  the  amount  of  divergence  will  follow  the  laws  of  probability.  For 
these  laws,  based  on  the  algebraic  law  expressing  the  number  of 
combinations  of  r  things  taken  n  at  a  time,  will  account  for  the  dif- 
ference between  the  constitution  of  a  total  series  and  the  constitution 
of  any  group  of  things  chosen  at  random  from  it,  consequently  for 
the  differences  between  any  two  measures  due  respectively  to  these 
two  constitutions. 

We  have,  consequently,  to  find  the  distribution  of  a  divergence 
(of  obtained  from  true  or  of  true  from  obtained)  and  know  before- 
hand, in  cases  of  random  sampling,  that  it  will  be  of  the  type  of  the 
probability  surfaces  given  in  Figs.  12  and  49,  will  be  symmetrical 
(since  the  true  is  as  likely  to  be  greater  as  to  be  less  than  the  ob- 
tained) with  its  mode  at  0  (since  all  that  we  do  know  about  the  true 
is  that  it  is  more  likely  to  be  the  obtained  measure  than  to  be  any 
other  one  measure).  What  we  need  to  know  is  its  form  and  vari- 
ability, to  know,  that  is,  how  often  we  may  expect  a  divergence  of 
.01,  how  often  one  of  .02,  how  often  one  of  .03,  etc.  Suppose  our 
obtained  measure  to  be  10.4  and  the  distribution  of  the  probable 
divergence  of  its  corresponding  true  measure  from  it  to  be  known  to 
be  as  follows  : 


—  1.1  to 

— 

.9 

—  .9  " 

— 

.7 

—  .7  " 

— 

.5 

—  .5  " 

— 

.3 

—  .3  " 

— 

.1 

—  .1  " 

+ 

.1 

+  -1  " 

+ 

.3 

+  .3  « 

+ 

.5 

+  .5  " 

■+ 

.7 

+  -7  " 

+ 

.9 

1  or   .01 

per  cent. 

10  "  1 

(i 

45  "  4.5 

u 

120  "  12 

a 

210  "  21 

a 

252  "  25 

n 

210 

120 

45 

10 

1 

+  .9   "    +1.1 

We  can  say  :  '  The  true  measure  will  not  rise  above  1  1.3 
(10.4  +  .9)  in  more  than  one  case  out  of  1,024/  or,  'The  chances 
are  over  1,000  to  1  against  the  measure  being  over  11.3,'  or,  '  The 
chances  are  nearly  99  to  1  against  the  true  measure  being  over  11.1/ 
or,  'The  chances  are  about  8  to  1  against  the  true  measure  differing 
from  10.4  either  above  or  below  by  more  than  .5.'* 

*  It  may  appear  strange  to  talk  about  the  true  measure,  which  is  a  fixed  value, 
'  rising  above'  or  'being  over,'  but  if  the  reader  will  bear  in  mind  that  we  do  not 
know  just  where  it  is  fixed,  but  do  know  the  probability  of  it^  being  at  this  or  that 
point,  he  will  not  misunderstand  the  terms  used.  They  could  not  well  be  avoided 
without  much  circumlocution. 


L38  MENTAL   AND  SOCIAL  MEASUREMENTS. 

If  the  form  of  the  distribution  of  the  divergence  were  known,  its 
variability  would  be  the  only  measure  needed.  The  form  will  always 
be  fairly  near  to  the  normal  surface  of  frequency  and  it  is  customary 
to  disregard  the  very  slight  error  involved  and  assume  the  form  to 
be  normal. 

If  we  know  the  variability  of  the  divergence,  the  probable 
frequency  of  any  divergence  or  of  divergences  less  than  or  greater 
than  any  given  amount  can  be  calculated  from  the  table  of  frequencies 
of  the  normal  probability  surface.  Conversely,  the  table  will  tell  us 
the  amount  of  divergence  which  will  be  exceeded  (or  not  exceeded) 
by  any  given  per  cent,  of  comparisons  of  true  and  obtained.  Illus- 
trations of  the  use  of  the  table  will  be  given  in  Chapter  XI. 

The  problem  of  determining  the  reliability  of  any  measure  due  to 
a  limited  series  of  samples  is,  then,  to  determine  the  variability  of 
the  fact,  divergence  of  true  from  obtained  measure.    (Ji"true  —  ^0bt.O 

It  is  clear  that  the  more  nearly  the  number  of  samples  taken  ap- 
proaches the  number  of  things  they  represent  the  closer  the  obtained 
measure  will,  in  general,  be  to  the  true  measure,  the  less  will  be  the 
range  of  divergence. 

It  is  clear  that  the  less  the  variability  amongst  the  individual 
samples,  the  less  will  be  the  divergence  of  the  obtained  from  the 
true  measure  of  central  tendency.  For  instance,  if  men  range  from 
4  to  7  feet  in  height,  averaging  5  feet  8  inches,  we  can  not  possibly 
get  an  average  more  than  1  foot  8  inches  wrong,  while  if  they  range 
from  2  to  10  feet,  we  may  make  an  error  of  3  feet  8  inches.  The 
same  holds  true  for  the  divergence  of  obtained  from  true  variability. 

Upon  these  facts  are  based  the  formulas  for  the  calculation  of  the 
variability  of  the  divergence  of  true  measure  from  that  obtained  from 
any  given  series  of  samples.  These  formulas  take  as  the  definition  of 
1  true  measure,'  the  measure  which  would  be  found  if  an  infinite  num- 
ber of  cases  were  studied. 

The  formulas  to  be  given  here  for  the  reliability  of  central  ten- 
dencies and  variabilities  are  those  in  common  use.  They  are  abso- 
lutely exact  only  for  a  case  where  the  distribution  of  the  trait  itself 
is  that  of  the  normal  probability  surface  with  extremes  at  minus 
infinity  and  plus  infinity,  and  so  are  never  absolutely  exact  for  any 
real  case.  They  are  very  inexact,  except  for  a  trait  showing  a  clear 
central  tendency  with  decreasing  frequencies  on  either  side.     This, 


THE  RELIABILITY  OF  MEASURES.  139 

however,  commonly  occurs  in  those  mental  measurements  from  which 
we  have  any  right,  according  to  the  principles  of  Chapters  III.  and 
IV.,  to  calculate  a  type  and  divergences  from  it.  The  A.  D.  and  a 
formulae  give  in  such  cases  a  variability  for  the  divergence  of  true 
from  obtained  that  is  a  trifle  too  large,  and  so  make  the  obtained 
result  seem  less  reliable  than  it  is.     This  is  perhaps  a  useful  error. 

The  Reliability  of  an  Average. 

The  probable  divergence  of  the  true  from  the  obtained  average? 
depending  upon  the  number  of  cases  and  the  variability  of  the  dis- 
tribution, may  be  calculated  according  to  different  formula?,  accord- 
ing as  we  use  0"dis.,  A.  D.dis  or  P.  E.dig  *  as  a  measure  of  the  variabil- 
ity of  the  distribution  from  which  the  average  was  obtained. 

If  we  use  0"dis.,  the  divergence  of  the  true  from  the  obtained  aver- 
age will  be  a  quantity  symmetrically  distributed  about  0  as  its  mode 
or  average,  with  a  variability  expressed  by  a  mean  square  deviation 
of  VdisjVn.     That  is,  au  av  _obt  av  =aAiJ\/n. 

Its  variability  in  terms  of  A.  D.  will  be  .7979<rdis jVn.  That 
is,  A.  D.t.  av._obt.  av.  =  .7979<7disViAi. 

Its  variability  in  terms  of  P.  E.  will  be  .6745*7  dis Jv/?i.  That  is, 
P-  E-t.  av.-obt.  av.  =  .6745<7dte.i/n. 

For  instance,  let  ^40bt  =  the  obtained  average  :  let  «7dig  =  the  vari- 
ability (mean  square  or  standard  deviation)  of  the  distribution  :  let 
At  =  the  average  that  would  be  obtained  from  an  infinite  number  of 
measures.  Then,  if  Aohl  =  20.2,  adis  4.2  and  the  number  of  meas- 
ures, 300,  At  —  AoU_  =  0  with  <rt_0  equal  to  4.2/17.32  or  .242,  Au  — 
Aljht  will  then  range  between  —  .726  and  +  .726  in  997  cases  out  of 
1,000,  between  —  .242  and  -f  .242  in  682  cases  out  of  1,000, 
between  —  .40  and  -f  .40  in  900  cases  out  of  1,000.  The  student 
can  verify  these  figures  from  the  table  on  page  148.  In  other  words, 
the  chances  are  997  to  3  or  332  to  1,  that  the  true  average  will  not 
deviate  from  the  obtained  by  more  than  .726  ;  682  to  318,  or  over  2 
to  1,  against  a  deviation  of  over  .242  ;  and  900  to  100,  or  9  to  1, 
against  a  deviation  of  over  .40.     In  still  different  words,  the  chances 

*  Since  to  measure  reliability  we  have  to  measure  the  variability  of  a  divergence 
and  shall  need  to  use  terms  similar  to  those  used  in  measuring  the  variability  of  in- 
dividual things  or  conditions,  it  will  be  well  to  name  the  average  deviation  of  a  dis- 
tribution of  a  thing  or  condition  A.  D.,n«..  Similarly,  a  and  P.  E.  in  the  sense 
hitherto  used  will  now  be  called  ff.iu.  and  I'.  E.,u».. 


140  MENTAL  AND  SOCIAL  MEASUREMENTS. 

are  2  to  1  that  the  true  average  lies  between  19.958  and  20.442  ;  9 
to  1  that  the  true  average  lies  between  19.8  and  20.6  ;  332  to  1  that 
the  true  average  lies  between  19.474  and  20.926. 

If  for  a  measure  of  the  original  distribution's  variability  we  take 
its  A.  r>.lUs..  the  variability  of  the  divergence  of  true  from  obtained 
average  will  be 

1.2533  A.  D.dis 

t.  av.  —  obt.  av. 


V'i 


4    D  — 

-"-•   -^U.  av.  —  obt.  av.  — 


A.   D.,iis. 


P.  E.,av._ 


obt.  av. 


.84435  A.  D.dig. 
Vn 


If  for  the  measure  of  the  original  distribution's  variability  we 
take  its  P.  E.dis  the  variability  of  the  divergence  of  true  from  ob- 
tained average  will  be 

_  1.4826  P.  E.di, 

t.  av.  — obt.  av.  _  /  — 

yn 

1.1843  P.  E.dI„ 
AD  — 

A"  ■L7't.  av.  —  obt.  av.  — 


P.  E.t. 


av.  —  obt.  av. 


yn 
P-  Edis. 


The  same  formulas  may  be  used  roughly  for  the  reliability  of  a 
median  if  the  student  himself  remembers  and  warns  his  readers  that 
the  divergence  of  true  from  obtained  median  may  exceed  the  amount 
shown  by  the  formulae.  Actually  the  excess  is  not  enough  to  lead  to 
serious  error. 

For  the  mode  too  the  same  formula?  may  be  used  as  a  rough  ap- 
proximation. In  proportion  as  the  mode  is  taken  to  cover  a  rela- 
tively wide  unit  the  formulas  will  give  too  great  apparent  unreliabil- 
ity. But  in  proportion  as  the  mode  is  assumed  on  the  mere  basis  of 
greatest  frequency  they  will  give  the  reverse. 

This  process  of  finding  the  probable  divergence  of  true  from  ob- 
tained measure  may  be  better  realized  by  testing  it  experimentally. 
For  example,  let  us  take   as   jT's  true  average  in  some   trait  the 


THE  RELIABILITY  OF  MEASURES.  141 

average  from  1,000  trials,  and  suppose  the   1,000  trials  to  be  dis- 
tributed as  follows : 

Quantity.        Frequency.  Quantity.        Frequency. 


10 

10 

11 

20 

12 

40 

13 

80 

14 

100 

15 

120 

16 

110 

17 

140 

18 

130 

19 

100 

20 

80 

21 

50 

22 

20 

T's  true  average  is  then  16.51. 

If  now  one  takes  1,000  discs  or  slips  of  paper  and  marks  10  of 
them  10,  20  of  them  11,  40  of  them  12,  etc.,  he  can  imitate  the  action 
of  random  selection  in  tests  by  random  drawings  from  the  discs.  If 
one  draws,  say  20,  and,  regarding  each  as  one  trial  in  the  tests,  com- 
putes the  average  of  the  20,  he  has  the  parallel  of  an  obtained 
average  from  N=  20.  The  patience  to  make  100  or  so  drawings 
of  20  or  so  each  will  be  rewarded  by  the  opportunity  to  distribute 
the  100  obtained  divergences  of  Av.true  from  Av.obt.  and  to  see  how 
far  this  distribution  conforms  to  that  obtained  from  the  formula? 
above. 

In  a  similar  experiment,  where  Av.true  was  2.5  and  27  drawings, 
each  of  20  from  a  series  of  400,  were  made,  the  actual  divergences 
were  as  given  in  column  I.  below.  The  probable  divergences  given 
by  the  average  of  the  27  formulae, 

p     ~p,  " •  1^-dis.  of  first  20 

■*•  •   -^n.  av.—  av.  obt.  from  first  20  = 


P.  E.t. 


av.— av.  obt.  from  second  20 


1/20 

X  .  X-/.(Jjg  0f  second  '. 

I   20 


etc.,  are  given  in  column  II.     The  P.  E.t_obt  from  experiment  is 
.212;  that  from  theory  is  .224. 


By  experi- 
ment.   By  theory. 

—  .G  arid  beyond  0  -9  + 
_.5to— .6                 1  .9  — 

—  .4 


—  .3 
2 

—  .1 
0 


—  .5  2  1.2  + 

—  .4  1  1.9  + 

—  .8  4  2.4  + 

—  .2  3  3.0  — 


By  experi- 

ment. 

By  theory 

Oto  +  .l 

3 

3.2 

+  .1  "  +-2 

3 

3.0  — 

+  .2  "  +.3 

5 

2.4  + 

+  .3  "  +.4 

1 

1.9  + 

+  .4" 

0 

1.2  + 

+  .5  "  +.6 

1 

.9  — 

3.2  +  .6  and  beyond  0  .9  + 


142  MENTAL  AND  SOCIAL  MEASUREMENTS. 

The  Reliability  of  a  Measure  of  Variability. 

As  before,  we  are  measuring  a  variable  fact,  'Divergence  of  true 
from  obtained  variability,'  which  has  a  mode  at  0,  the  distribution 
of  a  probability  surface,  and  a  variability  calculated  from  the  original 
series'  variability  and  number  of  cases. 

The  formulae  for  these  measures  of  variability  are  for  deviations 
from  the  average,  but  they  may  be  used  approximately  for  deviations 
from  the  mode  or  median. 

The  variability  of  the  divergence  of  the  true  variability  from  the 
obtained  variability  is  found  from  the  following  formulae  : 

<rdi,        1.2533  A.  D.di,       1.4826  P.  E.di, 
or r= —    -  or 


't.  var.-obt.  var.  —        /7r—    "A  /^—  ^L  /?J—  ) 


V2n  i/2n  i/2? 


<' 979*^        A.  D.dls        1.1843  P.  E. 


.  -|-v  Ul>.  Ul^ 

A.   D-t.  var.-obt.  var.  = 7^7=  ^  7s=~  °r 


\/2n  i/2n  l/2re! 

.G745,rdi,        .84435  A.  D.dis       P.  E.di, 

P-  E.t.va,_obt.var.  =  -  -  or -=-     -  or  . 

y2n  V2n  y2n 

The  Reliability  of  a  Measure  of  Difference. 

The  unreliability  of  a  difference,  say  between  Aohu  and  -Bobt.,  is 
measured  by  means  of  the  variability  of  the  divergence  between  the 
two  measures.  The  probable  true  measure  Au  is  distributed  about 
Aoht  as  a  mode  and  the  probable  true  measure  Bt  is  distributed  about 
B(jht  as  its  mode.  The  probable  true  difference,  that  is,  At  —  Bx , 
is  a  variable  with  its  mode  at  AoU  —  Boht  and  with  decreasing  fre- 
quencies as  we  take  ^40bt  —  Boht  -f  1,  Aoht  —  Boht  -f  2,  etc.,  or 
^obt.  -  ^obt.  -  1,  -4m.  -  ^obt.  -  2,  etc.  This  may  be  seen  most 
clearly  in  a  concrete  case  such  as  follows  : 

Given  the  facts  that  Aflht  =  42  and  Bohl  =  50,  that  the  differences 
between  Atrae  and  AoU  are  as  given  in  L,  and  the  differences  between 
i?true  and  Boht  are  as  given  in  II. 


I. 

II. 

Difference. 

Frequency. 

Frequency. 

—  2  to— 3 

1 

1 

—  1  "  —2 

5 

5 

0  "  —  1 

10 

10 

0  "  +1 

10 

10 

+  1  "  +2 

5 

5 

+  2  "  +3 

1 

1 

THE  RELIABILITY   OF  MEASURES. 


143 


To  find  the  difference  between  ^ttrue  and  Btrw.     From  I.  and  II. 


we  get  as  probable  values  of  Atme 

and  Btme,  III.  and  IV 

HI. 

IV. 

•^true- 

■Btrue- 

40  to  39 

1 

48  to  47               1 

41  "  40 

5 

49  "  48              5 

42  "  41 

10 

50  "  49             10 

42  "  43 

10 

50  "  51             10 

43  "  44 

5 

51  "  52              5 

44  "  45 

1 

52  "  53              1 

Using  for  each  distance  its  midpoint  value,  Atnie  and  Btrue  are 

-^true*  -Otrue* 

39.5  1  47.5  1 

40.5  5  48.5  5 

41.5  10  49.5  10 

42.5  10  50.5  10 

43.5  5  51.5  5 

44.5  1  52.5  1 


From  these  probable  values  of  Atrue  and  Btrne  we  get  the  follow- 
ing probable  differences  between  Atrue  and  Btrne: 


One     39.5 

with  one  47.5 

gives 

1  difference 

of 

8 

"     five  48.5s 

it 

5  differences 

it 

9 

"     ten  49.5s 

a 

10 

it 

a 

10 

"     ten  50.5s 

u 

10 

a 

a 

11 

"     five  51.5s 

a 

5 

it 

1 1 

12 

"     one  52.5 

a 

1  difference 

a 

13 

Five   40.5s 

with  one  47.5 

give 

5  differences 

it 

7 

"     five  48.5s 

it 

25 

1 1 

tt 

8 

"     ten  49.5s 

It 

50 

" 

tt 

9 

"     ten  50.5s 

a 

50 

a 

it 

10 

"     five  51.5s 

a 

25 

tt 

tt 

11 

"     one  52.5 

tt 

5 

tt 

a 

12 

Ten   41.5s 

with  one  47.5 

ti 

10  differences 

« 

6 

"    five  48.5s 

it 

50 

(< 

a 

7 

"     ten  49.5s 

it 

100 

(( 

it 

8 

"     ten  50.5s 

(( 

100 

tt 

tt 

9 

"     five  51.5s 

a 

50 

a 

tt 

10 

"    one  52.5 

u 

10 

" 

" 

12 

Ten    42.5s 

with  one  47.5 

(i 

K)  differences 

a 

5 

"    five  48.5s 

it 

50 

it 

tt 

(J 

"     ten  49.5s 

a 

100 

a 

" 

7 

"     ten  50.5s 

a 

100 

" 

" 

8 

"    five  51.58 

tt 

50 

(I 

" 

9 

"    one  52.5 

a 

10 

it 

" 

10 

1 1 1 


MENTAL  AND  SOCIAL  MEASUREMENTS. 


live  43. 5s  with  one  47.5 
"  five  48.5s 
"  "  ten  49.5s 
"  ten  50.5s 
"  five  51.5s 
<:  one  52.5 


gives      5  differ 

' '       25 

"       50 

"        50 

25 

5 


ences  of 


One    44.5 


with  one  -17.5 
"  five  48.5s 
"  ten  49.5s 
"  ten  50.5s 
"  five  51.5s 
"    one  52.5 


1  difference    " 

5  differences  " 
10 
10 

5 

1  difference    " 


Putting  together  all  these  differences  between  Atrue  and  Btme,  we 


Frequency. 

1  probable  difference   between  J-true  and  Bt 


have  : 

Quantity, 
of  3 
"  4 
"  5 
"  6 
"  7 
"  8 
"  9 
"  10 
"  11 
"  12 
"  13 

This  table  is  the  distribution  of  At     —  Bt    .     The  mode  is  —  8 

true  true 

[A  being  less  than  B)  or  Aoht  —  Boht    The  variability  is  P.  E.  =  1.12. 


10 

a 

differences         " 

45 

a 

120 

a 

210 

i< 

252 

c< 

210 

a 

120 

it 

45 

it 

10 
i 

(i 

tt 

riirmvonno                li 

Since  the  distribution  of  A, 


jfr     about  —  8  as  a  mode  is  the  same 


thing  as  the  distribution  of  the  divergence  of  Atme  —  Bt 
A0\)t. 


from 


J50bt  about  0  as  a  mode,  we  have 
P.  E., 


=  1.12. 


•(At  -BO-  (^obt— J?obt.) 

The  chances  are  1  to  1  that  the  true  difference  will  not  vary  from 
the  obtained  difference  by  more  than  1.12,  will  not  go  outside  of 
-6.88  and  -9.12. 

The  variability  of  the  divergence  between  the  true  measures  is 
thus  dependent  on  the  variabilities  of  the  divergences  of  each  one 
from  its  corresponding  obtained  measure.  The  unreliability  of  a  dif- 
ference between  two  measures  equals  in  fact  the  square  root  of  the 
sum  of  the  squares  of  the  unreliabilities  of  the  measures  themselves. 
The  formula  for  its  calculation  is,  Variability  of  (At  —  Bt) 


=  l/[var.  of  (Au  -  Jobt.)]2+  [var.  of  (Bu  -  Boht)  ] : 


THE  RELIABILITY  OF  MEASURES.  145 

Using  the  common  standards  of  measurement  of  variability, 


^diff.  At-Bu  =    "^(^t.-^obt.)2  +   (**t.-*obt.)J 


A.  D.dift,  At_Bt  =  V(A.T>.At_AohJ  +  (A.  B.St_BohJ 


P.  E.diff.  At  _  Bi  =  l/(P.  E.,t  _Aoht )2  +  (P.  E.Bt  _  BohJ 

The  most  probable  true  difference  is,  then,  the  obtained  differ- 
ence, and  the  chances  that  the  true  difference  is  so  much  less  or  so 
much  more  than  it  can  be  calculated  from  the  tables  for  the  proba- 
bility surface. 

TJie  Reliability  of  a  Pearson   Coefficient  of  Correlation. 

The  divergence,  in  a  case  of  lineal  correlation,  of  the  true  coeffi- 
cient of  correlation  from  that  obtained  from  the  limited  number  of 
pairs  of  measures  compared,  is  a  variable  trait  with  a  probable  mode 
at  0,  and  a  variability  which  serves  as  the  measure  of  the  unrelia- 
bility of  the  obtained  result.     The  formula?  *  are : 

1-r2 


A.  D 
P.  E. 


Vn(l  +  r2) 
.7979(1  -r2) 

l/w(l  +  r2) 
.6745(1  -r*) 


rt  —  i 


l/n(l  +  r2) 

It  is  customary  to  speak  of  the  variability  of  the  divergence 
of  true  from  obtained  measure  as  the  measure's  error.  Thus 
at.  av.-obt.av.  *s  called  the  mean  square  error  of  the  obtained  average  ; 
P.  E.t ..r._obt .  r.  is  called  the  probable  error  of  the  obtained  coefficient 
of  correlation;  A.  D.t  ,mr_obt  ,,nr  is  called  the  average  error  of  the 
obtained  difference.  These  terms  are  somewhat  ill  chosen,  as  there 
is  really  no  '  error,'  but  only  a  varying  degree  of  probable  approxima- 
tion.    I  have,  therefore,  used  the  word  unreliability  throughout. 

Problems. 

What  is  the  unreliability  of  each  of  the  averages  and  variabilities 
in  the  following  cases? 

*  There  is  some  uncertainty  about  these  formulae,  certain   authorities  favoring 
the  use  of  simply  }/n  in  the  denominators  in  place  of  ]Ai(l  +  r2). 
10 


lit;  MENTAL   AND  social    MEASUREMENTS. 


30. 

Av,,=  10. 

P. 

E>dla 

=  1. 

N  =  20. 

31. 

kv.s=  10. 

u 

"  1.5. 

"  "  30. 

32. 

Av.,.=  12. 

ii 

"  2.0. 

"  "  40. 

33. 

Av.jD=13. 

a 

"  3.0. 

"  "  40. 

3  I. 

Av.,=  14. 

a 

"  3.0. 

"  "  360. 

What  is  the  unreliability  of  each  of  the  following  differences  ?. 

35.  Av.6.  —  AvM  =  2.     The  data    concerning  Ax.A  and   Ay, 
being  as  in  30  and  32. 

36.  Av.jj  —  Ay.a  =  3.     The    data    concerning  Av.^  and  Av 
being  as  in  30  and  33. 

37.  Av.E  —  Ax.A  =  4.     The  data  concerning  AvM   and   Av, 
being  as  in  34  and  30. 

38.  Av.£  —  Av.B  =  4.     The    data  concerning  Ay.a   and  Av, 
being  as  in  34  and  31. 

39.  Av.£  —  Av.c=  2.     The    data   concerning  AvM  and  Av, 
being  as  in  34  and  32. 

What  is  the  unreliability  of  /■  in  each  of  the  following  cases  ? 

40.  r=  .46.     N=  200. 

41.  r  =  .16.     N=  200. 
-12.  r=  .16.     N=  600. 


CHAPTER  XL 

THE  USE  OF  TABLES  OF  FREQUENCY  OF  THE  PROBABILITY  SURFACE. 

Table  XLIII.  gives  for  any  normal  surface  of  frequency  the 
per  cent,  of  cases  included  between  the  average,  0,  and  any  degree 
of  deviation,  the  latter  being  measured  in  terms  of  the  standard 
deviation  of  the  distribution,  <7dis>.  Tables  XLIV.  and  XLV.  give 
the  same  information  when  the  degrees  of  deviation  are  in  terms  of 
the  A.  D.dig  and  P.  E.dis . 

Thus  the  first  line  of  entries  of  Table  XLIII.  reads  :  Between 
the  average  and  .01  a  either  above  or  below,  either  +  or  — ,  there 
are  .004  of  the  cases ;  between  the  average  and  +  S)'la  there  are 
.008  of  the  cases  ;  between  the  average  and  —  ,03<r  there  are  .0120 
of  the  cases,  etc. 

It  thus  enables  one  to  calculate  the  entire  distribution  of  any  trait 
which  is  normally  distributed,  the  average  and  variability  of  which 
are  known.  For  instance,  if  one  finds  for  discrimination  of  color 
that  the  average  =  24.0  and  the  standard  deviation  =  4.0,  one  finds 
from  the  table  that  the  ability  24  —  24.99  or  that  between  the  aver- 
age and  +  .25<r,  will  be  possessed  by  9.87  per  cent,  of  the  group  ; 
the  ability  24  —  25.99  or  that  between  0  and  +  .5<r  by  19.15  per 
cent.,  and  consequently  the  ability  25  —  25.99  by  19.15  —  9.87,  or 
9.28  per  cent.  By  thus  finding  the  percentages  included  between 
the  average  ability  and  different  amounts  of  deviation  from  it,  and  so 
between  any  two  given  limits  of  deviation  from  it,  one  gets,  as  the 
table  of  frequencies  in  our  illustrative  case,  Table  XLVI. 

This  use  of  the  tables  gives  a  convenient  means  of  measuring  the 
degree  to  which  the  measures  under  investigation  approximate  to  the 
probability  curve  distribution.  If  the  table  of  actual  frequencies  of 
the  measures  is  compared  entry  for  entry  with  the  frequencies  given 
for  corresponding  deviations  in  the  table  for  the  probability  curve, 
one  can  sec  at  a  glance  the  general  closeness  of  correspondence.  In 
making  such  comparisons  the  actual  frequencies  may  properly  be 
grouped  so  as  to  represent  only  18  or  more  grades,  and  any  most 
likely  central  point  may  be  chosen  with  which  to  make  the  central 
point  of  the  probability  surface  coincide. 

147 


148 


MEXTA/.    AM)   SOCIAL   MEASUREMENTS. 


For  example,  let  the  measures  in  the  first  column  of  frequencies 
of  Table  KLVII.  be  the  actual  distribution.  Their  average  is  76  —  ; 
their  median,  76  +  ;  and  their  most  likely  mode,  75  —  or  77  — .  76 
may  be  taken  as  the  central  point  for  the  comparison.  Their 
A.  D.afo  from  it  is  2.65  steps  (5.30  units).  In  the  second  column 
the  actual  frequencies  are  given  in  per  cents.     From  Table  XLIV. 

TABLE  XLIII. 

Table  op  Values  of  the  Normal  Probability  Integral  Corresponding 

to  Values  of  x/a  or  the  Fraction  of  the  Area  of  the  Curve 

Between  the  Limits  0  and  -(-  x/u  or  0  and  —  xja. 

Total  area  of  curve  assumed  to  be  10,000. 

x  =  deviation  from  mean. 

a  =  standard  deviation. 


X  <T 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

A 

0.0 

0000 

0040 

0080 

0120 

0160 

0200 

0239 

0279 

0319 

0359 

40 

0.1 

0399 

0438 

0478 

0517 

0557 

0597 

1  0636 

0676 

0715 

0754 

40 

0.2 

0793 

0832 

0871 

0910 

0948 

0987 

1026 

1064 

1103 

1141 

39 

0.3 

1179 

1217 

1255 

1293 

1330 

1368 

1406 

1443 

1480 

1517 

38 

0.4 

1554 

1591 

1628 

1664 

1700 

1737 

1773 

1808 

1844 

1879 

36 

0.5 

1915 

1950 

1985 

2020 

2054 

2089 

2124 

2157 

2191 

2225 

34 

0.6 

2258 

2291 

2324 

2357 

2389 

2422 

2454 

2486 

2518 

2549 

32 

0.7 

2581 

2612 

2643 

2672 

2704 

2734 

2764 

2794 

2823 

2853 

30 

0.8 

2882 

2910 

2939 

2967 

2995 

3023 

3051 

3078 

3106 

3133 

28 

0.9 

3160 

3186 

3212 

3238 

3264 

3290 

3315 

3340 

3365 

3389 

26 

1.0 

3414 

3438 

3461 

3485 

3509 

3532 

3555 

3577 

3600 

3622 

23 

1.1 

3644 

3665 

3686 

3708 

3729 

3750 

3770 

3791 

3811 

3830 

21 

1.2 

3850 

3869 

3888 

3906 

3925 

3944 

3962 

3980 

3997 

4015 

19 

1.3 

4032 

4049 

4066 

4083 

4099 

4115 

4132 

4147 

4162 

4178 

17 

1.4 

4193 

4208 

4222 

4237 

4251 

4265 

4279 

4292 

4306 

4319 

14 

1.5 

4332 

4345 

4358 

4370 

4383 

4395 

4406 

4418 

4429 

4441 

12 

1.6 

4452 

4463 

4474 

4485 

4496 

4506 

4516 

4526 

4536 

4545 

10 

1.7 

4554 

4564 

4573 

4582 

4591 

4600 

4608 

4617 

4625 

4633 

9 

1.8 

4641 

4648 

4656 

4664 

4671 

4678 

4686 

4693 

4700 

4706 

7 

1.9 

4713 

4720 

4726 

4732 

4738 

4744 

4750 

4756 

4762 

4767 

6 

2.0 

4773 

4778 

4783 

4788 

4794 

4799 

4804 

4808 

4813 

4817 

5 

2.1 

4822 

4826 

4830 

4834 

4838 

4842 

4846 

4850 

4854 

4858 

4 

2.2 

4861 

4865 

4868 

4872 

4875 

4878 

4881 

4884 

4887 

4890 

3 

2.3 

4893 

4896 

4899 

4901 

4904 

4906 

4909 

4911 

4914 

4916 

3 

2.4 

4918 

4921 

4923 

4925 

4927 

4929 

4931 

4933 

4935 

4936 

2 

2.5 

4938 

4940 

4942 

4943 

4945 

4946 

4947 

4949 

4951 

4952 

2 

2.6 

4953 

4955 

4956 

4958 

4959 

4960 

4961 

4962 

4964 

4965 

1 

2.7 

4966 

4967 

4968 

4969 

4970 

4970 

4971 

4972 

4973 

4974 

1 

2.8 

4975 

4975 

4976 

4977 

4978 

4978 

4979 

4980 

4981 

4981  ! 

0,5 

2.9 

4982 

4982 

4983 

4983 

4984 

4984 

4985 

4985 

4986 

4986 

0.5 

3 

4987 

4991 

4993 

4995 

4997 

4998 

4999 

4999  ; 

4999 

5000 

8 

5000 

1 

1 

THE   USE  OF  TABLES  OF  FREQUENCY.  149 

TABLE  XLIV. 

Table  of  Values  of  the  Normal  Probability  Integral  Corresponding  to 

Values  of  xj  (A.  D. ).     Total  Area  of  the  Surface  of 

Frequency  Taken  as  1,000. 

XI A.  D.  Multiples 
of  the  A.  D. 

0. 

1. 

2. 

3. 

4. 

TABLE  XLV. 

Table  of  Values  of  the  Probability  Integral  Corresponding  to  Values 

of  X/  (P.  E. ).     Total  Area  of  the  Surface  of 

Frequency  Taken  as  1,000. 


.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

000 

032 

063 

095 

125 

155 

184 

212 

238 

264 

288 

310 

331 

350 

368 

384 

399 

413 

425 

435 

445 

453 

460 

467 

472 

477 

481 

484 

487 

490 

492 

493.4 

494.6 

495.8 

496.7 

497.4 

498.0 

498.4 

498.7 

499.1 

499.3 

499.5 

499.6 

499.7 

499.8 

499.9 

500.0 

f.  t,.  .uumpi 
of  the  P.  E. 

es          .0 

.1 

.2 

.3 

.4 

.5 

.6            .7 

.8          .9 

0. 

000 

027 

054 

080 

106 

132 

157       182 

205       228 

,1. 

250 

271 

291 

310 

328 

344 

360       374 

388      400 

2. 

411 

422 

431 

440 

447 

454 

460       466 

471       475 

3. 

479 

482 

485 

487 

489 

491 

493       494 

495       496 

4. 

497 

498 

499 

5. 

499.7 

TABLE 

XLVI. 

Ability. 

Frequency 

IN 

Per  Cents., 

Av.  Being  24.0,  and  a  B 

EING  4.0. 

Ability. 

Frequency. 

Ability. 

Frequency 

Ability. 

Frequency. 

Less  than  11       0.06 

20- 

20.99 

6.80 

29-29.99 

3.88 

11-11.99 

0.07 

21 

8.19 

30 

2.68 

12 

0.17 

22 

9.28 

31 

1.73 

13 

0.32 

23 

9.87 

32 

1.05 

14 

0.60 

24 

9.87 

33 

0.60 

15 

1.05 

25 

9.28 

34 

0.32 

16 

1.73 

26 

8.19 

35 

0.17 

17 

2.68 

27 

6.80 

36 

0.07 

18 

3.88 

28 

6.30 

37  and  over 

0.0G 

19 

6.30 

are  calculated  the  frequencies  of  deviations  —  .5/2.65  to  -f-  .5/2.65, 
+  .5/2.65  to  +  1.5/2.65,  and  so  on.  These  are  given  in  the  third 
column  of  frequencies  of  Table  XLVII.  The  divergences  of  the 
actual  distribution  from  the  probability  curve  distribution  of  the 
same  central  point  and  variability  are  given  in  the  next  column,  and 
in  the  last  column  are  put  in  per  cents,  of  the  corresponding  prob- 
ability surface  frequencies.  Fig.  41,  on  page  50,  gives  (he  compari- 
son in  terms  of  space. 


150 


MENTAL    AM)   SOCIAL    M EASl'REMENTS. 


TABLE  XLVII. 

\.    CUAX    IhsTIMBUTION   OF    RATIO   OP   ATTENDANCE  TO   ENROLLMENT    IN 

Cities  of  U.  S.  Compared  with  Normal  Distribution. 


i. 

II. 

III. 

IV. 

V. 

Quantity. 

Actual 
Frequency. 

Actual           Frequency 
Frequency        in  normal 
in  per  cents.         surface. 

Differences. 

Differences 
in  per  cents. 
IV.  is  of  III 

45-46.9 

1 

.184 

+ 

.184 

? large 

7 
9 

1 
0 

.184 

,.025 

+ 

.184 
.05 

? large 
? large 

51 

2 

.37 

+ 

.35 

?  large 

3 

0 

.055 

— 

.055 

—  100 

5 

4 

.74 

.12 

+ 

.(32 

+    52 

7 

1 

.184 

.34 

— 

.156 

—    46 

9 

2 

.37 

.66 

— 

.29 

—    44 

61 

4 

.74 

1.3 

— 

.56 

—    43 

3 

15 

2.75 

2.4 

+ 

.35 

+    15 

5 

21 

3.85 

3.8 

+ 

.05 

+      1 

7 

34 

6.24 

5.9 

+ 

.34 

+      6 

9 

44 

8.07 

7.8 

+ 

.27 

+      3 

71 

31 

5.69              10.1 

— 

4.32 

—    43 

3 

54 

9.91               11.45 

— 

1.54 

—    13 

5 

65 

11.9                 11.9 

.000 

0 

7 

89 

16.3                11.45 

4-4.85 

+    42 

9 

70 

12.85               10.1 

+  2.75 

4-    27 

81 

37 

6.79 

7.8 

— 

1.01 

—    13 

3 

29 

5.32 

5.9 

— 

.58 

—    10 

5 

15 

2.75 

3.8 

— 

1.05 

—    28 

7 

11 

2.03 

2.4 

— 

.37 

—    15 

9 

9 

1.65 

1.3 

+ 

.35 

+    27 

91 

1 

.184 

.66 

— 

.376 

—    57 

3 

2 

.37 

.34 

+ 

.03 

+      9 

5 

1 

.184 

.12 

+ 

.064 

+    53 

7- 

98.9 

2 

.37 

.055 

+ 

.315 

+  573 

otal 

N  = 

545 

^.025 

? large 
? large 

To  find  the  frequency  of  any  given  ability  in  a  normal  distribu- 
tion, the  central  point  and  variability  of  which  are  known. 

The  frequency  of  any  degree  of  ability  can  obviously  be  calcu- 
lated quickly  if  the  average  and  variability  are  given.  For  instance, 
if  A  =  10  and  a  =  2.4,  how  many  cases  will  be  between  12.4  and 
12.6  ?  12.4  is  exactly  la  from  the  A  v.  and  12.6  is  1.0833<x  from 
the  Av.  The  per  cents,  of  cases  included  between  A  and  la  and 
between  A  and  1.08<r  are  respectively  34.14  and  36.00.  The  num- 
ber of  cases  between   la  and   1.08a  is  then   1.86   per  cent,   of  the 


THE    USE  OF  TABLES   OF  FREQUENCY. 


151 


whole  number  in  the  series.  To  be  exact  and  allow  for  the  .0033, 
we  add  to  the  last  figure  one  third  of  the  difference  in  the  table 
between  the  per  cents,  for  1.08  and  1.09,  viz.,  one  third  of  a  22  or 
.0007.  .3414  from  .3607  then  gives  us  .0193,  or  1.93  per  cent. 
The  number  of  cases  between  12.4  and  12.6  is,  then,  1.93  per  cent, 
of  the  whole  number  of  cases.  Practice  with  the  following  problems 
will  familarize  one  with  this  use  of  the  table  : 

43.  Av.  =  10.     a  =  3.     What  per  cent,  of  cases  lie  between  7 
and  13? 

44.  Av.  =  22.     a  =  4.4.     What  per  cent,  of  cases  lie  between  18 
and  20  ? 

a  =  2.1.    What  per  cent,  of  cases  lie  above  22  ? 

a  =  2.1.    What  per  cent,  of  cases  lie  below  13  ? 

A.  D.  =  3.46.     What  per  cent,  of  cases  lie 


45.  Av.  =  15.5. 

46.  Av.  =  15.5. 

47.  Av.  =  14.86. 
between  12  and  13? 

48.  Av.  =  14.86. 
between  14  and  16? 

49.  Av.  =  29.74. 
between  24  and  25  ? 


A.  D.  =  3.46.     Wliat  per  cent,  of  cases  lie 
P.  E.  =  3.18.     What  per  cent,  of  cases  lie 


To  find,  from  any  starting-point  on  the  scale  of  measurement,  the 
limits  of  ability  that  will  include  a  stated  percentage  of  the  cases. 

By  using  the  tables  the  other  way  about,  one  may  find,  Av.  and 
a  being  known,  the  degree  of  deviation  from  the  average  (or  the  dis- 
tance from  any  stated  point,  e.  g.,  the  upper  limit,  the  lower  limit, 
the  point  la  above,  etc.)  needed  to  include  any  stated  percentage  of 
the  cases. 

For  instance,  how  far  above  the  average  must  one  go  to  get  one 
fourth  of  the  cases,  the  Av.  being  8.0  and  a  2.0?  A  distance  of  .67<r 
includes  2,486  and  a  distance  of  .68<r  2,518.  A  distance  of  .675(7 
will  obviously  include  25  per  cent.,  .675  times  2  is  1.35.  Hence 
the  answer  is  9.35.  Again,  what  limits  of  ability  will  include  80 
per  cent,  of  the  cases  ?  From  knowledge  of  the  shape  of  the  normal 
surface  it  is  known  that  the  cases  are  thickest  the  nearer  they  are  to 
the  average.  So,  of  course,  we  take  in  the  example,  limits  equidis- 
tant from  the  average.  They  are  +  1.28«r  and—  1.28<x,  or  more 
exactly,  -f-  1.281  7t    and  —  1.281.7<r.      In  the  illustration  these  are 


L52  MENTAL  AND  SOCIAL  MEASUREMENTS. 

5.4366  and  10.5634.      In  reckoning  inward  from  either  extreme  it  is 
besl  to  arbitrarily  take  3<r  as  the  limit  pins  or  minus,  though  in  the 
theoretical  surface  the  limits  are  plus  infinity  and  minus  infinity. 
The  following  are  simple  problems  : 

50.  Av.  =  10  and  a  =  2.  What  limits  will  include  the  30  per 
cent,  just  above  the  average? 

51.  The  20  per  cent,  below  it? 

52.  The  middle  two  thirds  of  the  cases  ? 

53.  Av.  =  17.24.  A.  D.  =  4.6.  What  limits  will  include  the 
middle  three  fourths  of  the  cases  ? 

54.  The  bottom  10  per  cent.  ? 

55.  The  second  sixth  of  the  cases  from  the  top  ? 

This  use  of  the  tables  is  that  followed  in  transmuting  a  series  of 
measures  in  terms  of  relative  position  into  terms  of  amount.  In  so 
far  as  the  distribution  of  the  trait  is  that  of  the  probability  surface 
we  can,  calling  the  average  0,  find  the  limits  of  deviation  from  it  in 
terms  of  the  variability  as  a  unit  which  will  include,  say,  the  lowest 
1  per  cent.,  the  next  3  per  cent.,  the  8  per  cent,  from  the  23d  to 
31st  per  cent,  from  the  top,  etc.  The  process  is  so  far  identical  with 
that  in  the  examples  just  given.  Then  follows  the  calculation  of  an 
average  amount  to  fit  the  cases  included  between  each  pair  of  limits. 
How  this  is  done  may  be  seen  from  a  concrete  case.  Suppose  that 
of  400  boys'  themes  16,  or  4  per  cent.,  are  indistinguishable  for  ex- 
cellence, but  are  worse  than  100  and  better  than  284.  They  are 
then  per  cents.,  25,  26,  27  and  28.  By  Table  A  these  per  cents,  will 
lie  between  +  .6745<r  and  +  .5531<r.  By  the  table  we  find  that 
the  abilities  between  these  limits  have  the  following  frequencies  : 

Ability.  Frequency. 

.5531<rto.56<T  23 

.56  34 

.57  34 

.58  34 

.59  33 

.60  33 

.61  33 

.62  33 

.63  32 

.64  33 

.65  32 

.66  32 

.67ff  to  .6745ff  14 


THE   USE  OF  TABLES  OF  FREQUENCY.  153 

The  average  ability  for  the  group  is  .61  + .  This  was  the  method 
by  which  Tables  XXXI.  and  XXXII.  in  Chapter  VII.  were  con- 
structed. 

Given  the  unreliability  of  an  average  in  the  form  of  the  variabil- 
ity of  its  divergence  from  the  true  average  (<rt  Av._obt.  Av.  or  A. 
D-t.  Av.-obt.  av.  or  P'  E.t.  Av_obt  Av) ;  to  calculate  the  chances  that  the 
true  average  will  differ  from  the  obtained  by  any  given  amount.  The 
problem  is  simply  that  of  finding  the  frequency  of  any  degree  of  abil- 
ity in  a  normal  distribution  the  central  point  and  variability  of  which 
are  known. 

For  example,  at  Av.-obt.  av.  is  3.2.  To  find  the  chances  that  the 
true  average  will  not  vary  from  AoU  by  more  than  1.0,  2.0,  3.0,  4.0, 
6.0  and  10.0.  1.0  is  +  31  per  cent,  of  3.2.  By  the  table  deviations 
within  the  limits  +  .31<r  and  —  .31<r  occur  with  a  frequency  of 
12.17  +  12.17  or  24.34  per  cent.  There  is,  then,  1  chance  out  of 
4  that  AL  will  not  differ  from  AohL  by  more  than  1.0.  2.0  is  62  per 
cent,  of  3.2.  By  the  table  deviations  within  the  limits  +  .62<r  and 
—  .62<t  occur  in  45.8  per  cent,  of  the  cases.  The  chances  are  almost 
1  to  1  that  At  will  not  differ  from  Aoht  by  more  than  2.0.  The 
chances  of  a  difference  of  less  than  10  will  be  found  to  be  9,986 
out  of  10,000,  or  over  700  to  1. 

Given  the  unreliability  of  Aoht  in  the  same  way  as  above,  to  cal- 
culate the  amount  of  divergence  of  Au  from  Aoht  more  than  which 
has  a  given  degree  of  improbability. 

This  problem,  the  converse  of  the  above,  is  identical  with  that  of 
calculating  limits  of  ability  from  the  average  as  a  starting-point. 

For  example,  at  av.-obt.  av.  IS  ^.0.  To  find  the  amount  of  differ- 
ence between  At  and  A„ht ,  differences  greater  than  which  will  have 
only  1  chance  in  100  of  happening.  In  the  table  we  find  the  dis- 
tance from  the  average  which  must  be  passed  over  in  both  pins  and 
minus  directions  to  include  99  out  of  100  cases,  49.5  plus  and  49.5 
minus.  It  is  2.575tf.  Since  a  equals  3.0  the  answer  to  our  problem 
is  7.725. 

It  will  be  noted  that  the  tables  serve  equally  well  in  the  many 
cases  where  the  desired  fact  is  the  probability  of  a  given  divergence 


1">4  MQNTAL   AND  SOCIAL   MEASUREMENTS. 

Ay.  =  10  in  one  direction  or  the  amount  of  divergence  in  one  direc- 
tion, more  divergence  than  which  has  a  given  degree  of  improbability. 

The  same  methods  serve  if  the  unreliability  is  of  a  variability  or 
of  a  difference  or  of  a  relationship  —  in  short,  for  all  cases  where  the 
unreliability  is  measured  by  the  variability  of  a  divergence  of  true 
from  obtained,  and  this  divergence  is  distributed  in  a  normal  prob- 
ability surface. 

The  following  problems  will  offer  opportunity  for  acquiring  self- 
confidence  in  the  use  of  the  tables  in  connection  with  all  sorts  of  ques- 
tions about  unreliability  : 

56.  o't-o.Av.  =  1.6.  («)  What  is  the  probability  of  a  difference 
between  Av.t  and  Av.0  of  4.0  or  more?  (b)  What  are  the  chances 
that  Av.t  will  be  3.2  greater  than  Av.0  ?  (c)  Between  what  limits 
will  the  true  average  lie  with  a  probability  of  9999  to  1  ? 

57.  ox  _0  var  =  .4.  (a)  What  is  the  probability  that  the  true 
variability  is  more  than  .8  less  than  the  obtained?  (6)  That  the 
true  variability  is  not  more  than  .6  above  or  below  the  obtained  ? 

58.  <7t._0.  tuff.  = -5.  The  actually  obtained  difference  is,  Av.,  - 
Av.2  =1.2.  (a)  What  is  the  probability  that  the  true  difference  is  zero 
or  less  than  zero  ?  (6)  That  the  true  difference  is  :  Avn  —  Av.2 
=  2.4  or  more?  (c)  That  the  true  superiority  of  Av.x  over  Av.2  is 
between  1.7  and  .7  ?  (rf)  What  limits  would  you  assign  for  the  true 
difference  to  be  sure  that  the  chances  would  be  20  to  1  against  their 
being  exceeded  ? 

59.  r0  =  -f  .48.  at  _0  rel  =  .04.  (a)  Between  what  limits  does 
the  true  relationship  lie  with  practical  certainty  (it  is  customary  to 
take  997  out  of  1,000  as  practical  certainty)  ?  (6)  What  is  the 
chance  that  the  true  relationship  is  as  low  as  .40  ? 

60.  Av.0.  =  22.6.  A.  D.t._0.Av.  =  .4.  (a)  What  is  the  chance 
that  the  true  average  is  as  large  as  24.0  ?  (6)  That  it  is  as  small 
as  22.0? 

61.  Av.0.  =  28.2.  P.  E.t._0.Av.  =  .6.  (a)  What  is  the  chance 
that  the  true  average  is  less  than  26.0?  (6)  That  it  varies  from 
Av.0  by  less  than  2.0  ? 

62.  If  it  were  true  that  the  chances  were  82  to  18  that  the  true 
average  would  not  vary  from  the  obtained  by  more  than  13.4,  what 
would  be  the  value  of  P.  E.t  _0  Av  ? 


THE   USE  OF  TABLES  OF  FREQUENCY.  155 

63.  Av.1=10.1,  Av.2=12.4.  P.  E.t._0.„Av.iaudAv-2=  1.0. 
(a)  What  are  the  chances  that  Av.j  —  Av.2  =  0  or  less?  (6)  1.0  or 
less?  (c)  2.5  or  more?  (d)  Between  2.0  and  2.8?  (e)  Between  1.0 
and  3.3? 

64.  P.  E.disobt.  =  1.6,  A.  D.t._0.vai,  =  0.1.  («)  What  are  the 
chances  that  P.  E.dis  will  be  between  1.4  and  1.8  ?  (6)  That  it  will 
not  exceed  1.9?  (c)  What  limits  must  be  taken  such  that  the  true 
P.  E.(lis  will  be  practically  certain  (see  question  59)  not  to  exceed 
them? 

65.  ra  =  4-  .39,  P.  E.t_ft  reL  =  .008.  What  is  the  chance  of  the 
true  relationship  being  as  high  as  4-  40  ?  As  -f  41  ?  As  -f  .42  ? 
As  +  .50? 

66.  Speaking  roughly,  the  true  measure  is  practically  certain  to 
lie  between  the  following  limits : 

Obtained  measure  +  3<?t.—o.  measure  and  obtained  measure  — 3fft._0.  measure. 

"       +  3|  A.  D.t.— o.  measure  and  obtained  measure  —  3|  A.  D.t.—0.  measure. 

(t  H  I       Al    T>     T7>  ((  it  it  A\    T>      T71 

"T  ^-1  Jt.  -"-t. —  o.  measure  — tj  *•   -^-t. —  o.  measure- 

Justify  this  statement  from  the  tables. 

67.  rlo  -  r2o  =  .04,  P.  E.t._0.fliff.nand,.2  =  .06.  (a)  What  is  the 
chance  that  the  true  r2  is  really  equal  to  or  greater  than  the  true  r,? 
(6)  What  is  the  chance  that  the  true  r\  is  greater  than  the  true  r.,? 


Given  the  fact  that  two  groups  are  normally  distributed  and  that 
the  central  tendency  of  the  first  is  X  plus  the  central  tendency  of 
the  second,  X  being  in  terms  of  the  variability  of  the  first,  what  per 
cent,  of  the  first  group  will  exceed  the  central  point  for  the  second  ? 
The  per  cent,  will  equal  50  plus  the  per  cent,  included  between  the 
central  point  and  a  point  X  above  it.  (See  Fig.  86.)  This  is,  of 
course,  given  directly  by  the  table.     For  instance,  let  group  1   have 


156  MENTAL   AND  social   MEASUREMENTS. 

and  ffajg  =  4.  Let  group  2  have  Av.  =  8.  The  difference  -f  2 
equals  .5<r  (of  distribution  of  group  1).  The  percentage  of  group  1 
exceeding  the  average  for  group  2  will  be  50  +  19.15  or  69.15  per 
cent. 

"When  the  first  group  is  inferior  to  the  second,  the  calculation  is 
the  same,  replacing  50  per  cent,  plus  by  50  per  cent,  minus. 

68.  If  boys  in  spelling  average  18.6  with  <rdis  =  2.4,  and  girls 
average  20.0,  what  per  cent,  of  boys  will  reach  or  exceed  the  average 
for  girls  ? 

69.  If  the  per  cent,  of  attendance  to  enrollment  in  cities  averages 
74  with  a  P.  E.dis  of  8.6,  and  the  same  trait  in  towns  averages  64, 
what  per  cent,  of  cities  will  reach  or  exceed  the  average  for  towns  ? 

70.  If  the  median  strength  of  10-year-old  boys  is  16.2  with 
am  =2.1,  and  the  median  strength  of  11 -year-old  boys  is  17.4, 
what  per  cent,  of  10-year-olds  will  be  stronger  than  the  median 
11 -year-olds? 


CHAPTER   XII. 

SOURCES    OF    ERROR    IN    MEASUREMENTS. 

So  far  our  supposition  has  been  that  the  measures  with  which 
we  start  are  accurate  representatives  of  the  fact  measured,  that  A 
really  did  misspell  the  word  which  we  score  misspelled,  that  B  did 
really  take  the  .150  sec.  to  react  which  the  chronoscope  recorded, 
that  the  school  eurollment  and  average  attendance  given  for  cities  in 
the  U.  S.  Commissioner's  report  give  the  real  facts,  that  the  number 
of  children  recorded  in  certain  genealogy  books  for  certain  families 
were  the  real  numbers.  Our  problem  has  been  to  make  the  best  use 
of  the  data  and  introduce  no  error  in  manipulating  them.  But  that 
a  measure  should  thus  perfectly  represent  a  fact,  the  fact  must  be 
measured  by  a  perfect  instrument  used  by  an  infallible  observer.  In 
reality,  any  measure  is  a  compound  of  a  fact  and  the  errors  which 
the  instrument  and  observer  will  surely  make. 

These  errors  may  be  constant  or  variable.  A  constant  error  is 
one  tending  more  in  one  direction  than  the  other.  A  watch  that  is 
too  slow,  a  tendency  of  school  superintendents  to  make  the  attend- 
ance record  too  high,  are  examples.  Variable  or  chance  errors  are 
those  tending  in  the  long  run  to  make  the  amount  lower  as  often 
and  as  much  as  higher.  The  unevenness  in  action  of  a  delicate 
balance  due  to  dust,  air  currents,  etc.,  the  errors  in  addition  made 
by  the  clerks  in  a  superintendent's  office,  are  examples. 

Variable  errors  do  not  make  any  measure  unfair,  but  only  less 
exact  and  less  reliable.  If  a  body  is  weighed  by  an  instrument  which 
fluctuates  so  as  to  give  15(3.1,  156.2,  156.3,  156.3,  156.3,  L56.3, 
156.4,  156.4  and  156.4  in  nine  measurements,  but  is  known  not  to 
weigh  too  light  or  heavy,  156.3  is  a  true  measure,  but  the  L56.3 
only  means  between  15(5.25  and  156.35  and  there  is  a  slight  chance 
of  its  being  156.2  or  156.4  (about  1  chance  in  500). 

If,  on  the  contrary,  a  body  is  weighed  by  an  instrument  which 
fluctuates  so  little  as  to  give  156.298,  L56.299,  156.300,  15(1.300, 
L56.300,  L56.301,  156.301  and  156.301,  and  which  is  known  not 
to  weigh  too  light  or  heavy,  the  156.300  means  between  L56.2995 
and  156.3005  and  there  is  now  certainty  that  the  measure  is  not  so 

157 


158  MENTAL   AND  SOCIAL  MEASUREMENTS. 

low  as  156.2  or  so  high  as  156.4.  Indeed,  there  is  certainty  that  it 
is  between  L56.298  and  156.302. 

There  is  no  great  advantage  in  decreasing  the  amount  of  the 
variable  error  by  using  more  delicate  instruments  or  more  care  in 
observing,  unless  the  precision  and  reliability  thereby  obtained  can 
be  preserved  in  the  further  use  of  the  measurements.  The  advan- 
tage that  there  is  consists  in  the  moral  and  intellectual  training  one 
gets  and  in  the  possibility  that  the  measures  may  later  be  used  for 
purposes  other  than' one  expects. 

If  we  wish  to  get  A's  average  error  in  trying  to  equal  a  100-mm. 
line,  measurements  may  be  made  with  the  aid  of  a  glass  to  -^  mm., 
but  the  variation  between  A's  separate  trials  is  so  great  that  the 
larger  error  due  to  measuring  each  line  so  roughly  as  into  ^  mms.  is 
insignificant.  Indeed,  measurements  to  a  millimeter  really  do  as 
well.  If  we  wish  to  compare  the  reaction  time  of  1,000  boys  with 
that  of  1,000  girls,  the  median  of  10  times  being  taken  for  each 
individual,  measures  in  hundredths  of  seconds  will  do  as  well  as 
measurements  in  thousandths. 

Much  time  may  be  wasted  in  refining  measurements  in  cases 
where  no  advantage  accrues.  And  much  ignorance  is  shown  by  the 
many  students  who  disparage  all  measurements  that  are  subject  to  a 
large  variable  error.  They  either  do  not  know  or  forget  that  the 
reliability  of  a  measure  is  due  to  the  number  of  cases  as  well  as  to 
their  variability,  and  that  in  the  more  complex  and  subtle  mental 
traits  it  is  always  practicable  to  increase  the  number  of  measure- 
ments, but  often  impossible  to  make  them  less  subject  to  variable 
errors.  They  also  forget  that  the  natural  and  real  variability  of  the 
fact  itself  is  often  so  large  as  to  make  the  variability  due  to  errors 
of  instruments  and  observation  practically  negligible. 

Constant  errors,  on  the  other  hand,  are  never  negligible. 

The  errors  we  make  in  interpreting  handwriting  would  not,  in  a 
comparison  of  1,000  boys  with  1,000  girls  in  spelling  ability,  be 
worth  spending  a  day  on,  even  if  thereby  one  could  rectify  them  all, 
but  if  the  teachers  of  the  girls  pronounced  the  words  more  clearly 
and  phonetically  than  those  of  the  boys,  it  would  be  necessary  to 
discuss  the  proper  discount  or  give  up  all  hopes  of  precision.  That 
a  genealogist  by  mistake  sometimes  writes  4  or  7  matters  practically 
nil  to  the  student  of  vital  statistics,  but  the  genealogist's  constant 


SOURCES  OF  ERROR  IN  MEASUREMENTS.  159 

tendency  to  omit  more  children  than  he  adds  because  of  the  difficulty 
of  getting  complete  family  records,  is  of  the  utmost  importance. 

Increasing  the  number  of  measures  has  here  no  beneficial  in- 
fluence. In  certain  cases  increasing  the  number  of  observers  may, 
namely,  when  the  constant  error  of  one  observer  is  offset  by  the  con- 
stant error  in  the  opjjosite  direction  of  another  observer.  If,  that  is, 
there  is  an  error  of  prejudice  or  tendency  constant  for  any  one  ob- 
server, but  varying  in  direction  by  chance  among  a  group  of  observ- 
ers, what  is  a  constant  error  for  one  becomes  a  variable  error  for  a 
group,  and  is  no  longer  a  source  of  misleading,  but  only  of  lessened 
reliability.  For  instance,  if  any  one  person,  even  an  expert  judge, 
should  rank  100  men  in  order  for  morality  or  efficiency  or  intellect, 
the  results  would  probably  have  a  constant  error  due  to  the  undue 
weight  he  would  put  upon  certain  evidence ;  but  if  we  took  the 
median  of  the  rankings  given  by  ten  or  twelve  expert  judges,  the 
error  would  in  the  main  be  only  a  chance  error,  for  the  prejudice  of 
one  would  offset  the  prejudice  of  another. 

The  sources  of  constant  errors  in  mental  measurements  are  so 
numerous  and  so  specialized  for  different  kinds  of  facts  that  it  is  im- 
possible to  forearm  the  student  against  them  here.  Skill  in  avoid- 
ing them  is  due  to  capacity  and  watchfulness  far  more  than  to 
knowledge  of  any  formal  rules.  It  is,  however,  practically  wise  to 
test  any  result  which  may  be  affected  by  some  constant  error  by  using 
different  methods  of  measurement,  and  to  examine  the  means  of 
selecting  cases  for  measurement  with  the  utmost  care.  The  tendency 
to  bias  or  to  blunder  is  much  more  likely  to  make  one  select  unfair 
cases  than  to  make  one  measure  them  unfairly. 

There  is  also  a  source  of  error  which  is  perhaps  in  strictness  an 
error  in  inference,  but  which  from  another  point  of  view  may  be  re- 
garded as  an  error  in  measurement  and  so  as  relevant  to  the  topics  of 
this  book.  In  measuring,  say  the  spelling  ability  of  a  number  of  in- 
dividuals whom  we  wish  to  compare,  we  assume  that  the  achieve- 
ment of  each  is  a  measure  of  the  spelling  ability  of  each.  But  A  and 
JB  may  have  been  seated  where  they  did  not  hear  the  words  pro- 
nounced so  well  as  did  ( 'and  I).  E  and  F  may  have  had  headaches, 
while  G  and  // were  cheerful  and  bright.  There  exist  errors  due  in 
the  first  example  to  outer  physical  conditions  and  in  the  second  to 
inner  or  psychological   conditions.     To  compare  A,   B,    C,  etc.,  in 


160  MENTAL  AND  SOCIAL  MEASUREMENTS. 

spelling  ability,  every  extrinsic  condition  influencing  that  ability 
should  be  alike  for  all.  Otherwise  we  are  led  into  errors,  which  may 
be  called  errors  of  inferring  au  ability  in  abstracto  from  its  manifes- 
tation under  particular  conditions,  or  of  measuring  a  fact  with  a  con- 
stant error  of  condition.  It  will  be  simpler  to  treat  separately  errors 
due  to  physical  conditions  and  errors  due  to  mental  conditions. 

Errors  due  to  physical  conditions  can  be  prevented  by  making 
the  conditions  identical,  or  turned  into  relatively  harmless  variable 
errors  by  measuring  each  individual  a  number  of  times  under  condi- 
tions chosen  at  random.  It  would  seem  at  first  sight  best  to  make 
conditions  identical  wherever  practicable.  This  rule  probably  does 
hold  for  physical  measurements,  but  there  are  certain  disadvantages 
in  this  procedure  in  mental  measurements.  Too  much  artificiality 
and  restraint  in  conditions  often  lead  to  an  unusual  and  perturbed 
state  of  mind  in  the  person  measured,  such  that  the  thing  one  meas- 
ures is  likely  to  be  a  thing  which  would  never  occur  in  the  ordinary 
course  of  the  person's  life.  Measuring  precisely  a  fact  which  you  do 
not  want  is  worse  than  measuring  inexactly  the  fact  you  do  want. 

For  instance,  measurements  of  spelling  under  the  unequal  condi- 
tions of  a  schoolroom  would,  in  spite  of  them,  be  better  than  measure- 
ments from  10-year-olds  made  to  stand  one  at  a  time  in  the  sound- 
proof room  of  a  laboratory  with  head  exactly  50  centimeters  from  a 
phonograph  wrhich  pronounced  the  words  for  them  to  spell.  The  last 
method  would  give  identity  of  physical  conditions,  but  would  meas- 
ure insensibility  to  strange  surroundings  and  treatment  and  ability 
to  attend  to  and  interpret  the  phonograph's  noises  perhaps  more  than 
it  would  spelling  ability. 

Errors  due  to  mental  conditions  can  not  be  prevented  with  surety 
by  making  the  conditions  identical,  for  it  is  not  in  the  power  of  the 
observer  to  control  the  mental  conditions  of  the  person  measured. 
The  best  that  can  be  done  is  to  avoid  any  probable  cause  of  differ- 
ence in  them  and  to  take  the  subjects'  reports  as  to  what  their  men- 
tal conditions  are.  But  mental  conditions  vary  greatly  even  despite 
the  apparent  absence  of  causes  for  difference  ;  and  the  reports  of  mental 
condition  from  untrained  self-observers  must  be  vague,  subject  to 
constant  errors  and  always  from  a  personal  standard  of  comparison 
incommensurate  writh  that  of  any  other  individual.  Though  A  say.-, 
'  I  am  tired,'  and   B  says,  '  I  am   not,'  their  feelings  of  fatigue  may 


SOURCES  OF  ERROR  IN  MEASUREMENTS.  161 

be  equal.  We  do  not  take  untrained  individuals'  opinions  as  facts 
elsewhere  in  science,  and  have  no  right  to  do  so  here.  The  more 
reliable  procedure  would  be  to  eliminate  the  influence  of  the  variabil- 
ity of  inner  conditions  by  random  choice  from  among  them  rather 
than  to  pretend  to  eliminate  the  variation  itself. 

It  is  also  a  fair  question  whether  the  attempt  to  make  all  the 
mental  conditions  except  the  one  to  be  measured  alike  in  the  persons 
to  be  compared,  does  not  commonly  result  in  so  much  unnaturalness 
of  the  sort  against  which  protest  was  made  a  page  back,  as  to  do  more 
harm  than  good.  Attempted  restriction  of  mental  conditions  surely 
disturbs  any  one  even  more  than  restriction  of  physical  conditions. 

Success  in  eliminating  disturbing  conditions  is  not  attainable  as  a 
result  of  knowledge  of  any  fixed  rules,  but  only  through  a  happy  in- 
genuity in  devising  experiments,  arranging  observations  and  selecting 
data.  We  can,  however,  be  careful,  after  securing  the  best  measure- 
ments that  we  can,  to  distinguish  sharply  between  the  actual  meas- 
urement of  the  fact  under  certain  conditions,  on  the  one  hand,  and  on 
the  other  the  inferences  that  we  may  be  tempted  to  make  about  the 
fact  in  general  or  apart  from  those  particular  conditions.  It  is  not 
undesirable  to  make  inferences,  but  it  is  highly  undesirable  to  con- 
fuse them  with  measurements  or  to  leave  them  without  critical 
scrutiny. 

Much  more  might  well  be  said  with  regard  to  the  sources  of  error 
prevalent  in  studies  of  human  nature,  but  the  proper  bounds  of  an 
introduction,  not  to  the  logic  or  general  method  of  the  mental  sciences, 
but  only  to  their  statistical  problems,  have  already  been  passed. 

Weighting  Results. 

Different  sources  of  information  concerning  any  one  quantity  may 
give  it  differing  amounts,  and  these  sources  may  be  of  unequal  reli- 
ability. It  is,  then,  desirable  to  allow  more  weight  to  the  more 
trustworthy  sources  in  deciding  what  amount  is  the  most  probable 
for  the  quantity.  For  instance,  if  an  expert  in  physical  anthropology 
measured  A' a  head  and  scored  his  cephalic  index  .81,  while  an  ordi- 
nary person  scored  it  .80,  we  should  choose  the  .81  rather  than  the 
.80,  and,  if  we  allowed  something  for  each  judgment,  would  perhaps 
take  80.8  as  the  figure,  counting  the  anthropologist's  result  four  times. 

No  care  in  weighting  sources  will  do  so  much  service  as  the 
11 


162  MENTAL  AND  SOCIAL   MEASUREMENTS. 

elimination  of  constant  errors ;  and  ideally  no  source  with  a  constant 
error  unallowed  for  should  have  any  place  in  determining  a  result. 
Any  source  may  deserve  weight  because  of  either  numerical  or 
qualitative  strength.  Its  numerical  strength  is  as  the  square  root  of 
the  number  of  cases  whose  study  it  represents.  Weighting  for  quality 
is  bound  in  practice  to  be  largely  arbitrary,  but  this  is  not  a  great 
misfortune,  for  the  result  will  rarely  be  altered  appreciably  by  such 
differences  in  the  system  of  weighting  as  reasonably  competent 
students  would  make.  For  instance,  A,  B  and  C  with  the  same 
general  problem  use  different  methods  and  get  as  a  certain  correla- 
tion coefficient  .60,  .50  and  .48  respectively.  Suppose  that  we 
weight  these  sources  1,  1,  and  1  ;  4,  4  and  5 ;  3,  4  and  5  ;  and 
finally  4,  3  and  5.  We  have  then,  as  the  probable  true  coefficient, 
.5267,  .5231,  .5167  or  .5250.  Bowley  gives  a  rule  that  is  satisfac- 
tory for  most  cases  that  occur  in  practice,  namely,  to  give  your  atten- 
tion to  eliminating  constant  errors  and  not  to  manipulating  weights.* 
If  results  are  weighted  it  is  always  well  to  give  them  in  their  un- 
weighted form  as  well  and  leave  the  opportunity  open  for  any  critic 
to  weight  them  as  he  judges  proper. 

*  'In  calculating  averages  give  all  your  care  to  making  the  items  free  from  bias 
and  leave  the  weights  to  take  care  of  themselves.'      '  Elements  of  Statistics,'  p.  118. 


CHAPTER  XIII. 

CONCLUSION.   REFERENCES  FOR  FURTHER  STUDY. 

I  trust  that  the  reader  has  been  impressed  by  now  with  the  fact 
that  the  theory  of  mental  measurements  is  no  display  of  mathematical 
pedantry  or  subtle  juggling  with  figures,  but  on  the  contrary  is  simple 
common  sense.  The  chief  lessons  of  this  book  are  in  fact  simple  ap- 
plications of  the  most  elementary  logic.  They  may  be  summed  up 
in  the  form  of  warnings  against  certain  fallacies  common  in  the  quan- 
titative treatment  of  mental  facts,  viz. : 

1.  Accepting  guessed  equality  or  mere  verbal  likeness  in  place 
of  real  equality. 

2.  Using  quantities  on  a  scale  without  consideration  of  the  mean- 
ing of  the  scale's  zero  point. 

3.  Dealing  carelessly  with  totals  the  constitution  of  which  is 
unknown. 

4.  Using  an  average  to  represent  a  series  of  individual  measures 
regardless  of  their  distribution. 

5.  Estimating  a  total  series  from  individual  measures  numerically 
insufficient  or  so  selected  as  to  actually  misrepresent  it. 

6.  Estimating  differences  by  ambiguous  measures. 

7.  Using  a  difference  between  or  change  in  averages  to  represent  a 
series  of  individual  differences  or  changes.  (7  is  essentially  the  same 
fallacy  as  4.) 

If  the  reader  has  been  rendered  immune  to  these  errors,  has  ac- 
quired facility  and  confidence  in  the  manipulation  of  measurements, 
and  has  learned  to  discard  guess  work  and  crude  arithmetic  in  favor 
of  accurate  and  modern  methods  of  measuring  facts  and  relationships, 
the  purpose  of  this  book  has  been  amply  fulfilled. 

It  is  desirable  that  the  student  who  has  been  thus  introduced  to 
statistical  methods  should  proceed  to  study  samples  of  their  concrete 
application  to  problems  in  the  mental  sciences  and,  in  case  he  has  the 
necessary  mathematical  interest  and  training,  that  he  should  study  the 
abstract  properties  of  different  types  of  distribution,  the  derivation  of 
statistical  formulae,  the  mathematical  theory  of  correlation  and  other 

163 


1G4 


MENTAL   AND  SOCIAL  MEASUREMENTS. 


topics  in  pure  statistics.     To  these  cuds  the  references  given  below 
may  be  useful. 

These  will  be  grouped  in  accordance  with  the  different  interests 
which  may  be  supposed  to  dominate  the  quantitative  studies  of 
readers  of  this  introduction,  under  psychology,  education,  economics 
and  social  science,  anthropometry,  vital  statistics  and  biology.  A 
few  references  to  the  most  easily  understood  articles  on  pure  statistics 
will  form  a  group  by  themselves.  The  order  in  which  the  references 
for  each  topic  are  given  is  that  in  which  the  student  may  profitably 
read  them. 

Psychology. 

'  On  the  Perception  of  Small  Differences.'     By  G.  S.  Fullerton  and 
and  J.  McK.  Cattell.     No.  2  of  the  Philosophical  Series  of  the 
Publications  of  the  University  of  Pennsylvania,  May,  1892.    The 
University  of  Pennsylvania  Press,  Philadelphia. 
Quantitative  exactitude  was  first  sought  by  psychologists  in  the 
case  of  the  ability  to  perceive  differences.     The  monograph  by  Ful- 
lerton and  Cattell  gives  a  clear  account  of  the  common  methods 
of  estimating  quantitatively  psycho-physical  relationships,  viz.,  the 
method  of  the  just  noticeable  difference,  the  method  of  right  and 
wrong  cases,  the  method  of  average  error  and  the  method  of  mean 
gradation.     It  also  represents  an  investigation  made  with  full  con- 
sciousness and  appreciation  of  the  special  problems  of  variable  phe- 
nomena.    It  is  thus  the  best  introduction  to  the  special  problems  in 
mental  measurement  which  confront  the  student  of  psycho-physics. 

Table  for  Determining  the  Probable  Error   From  the  Percentage   of 
Eight  Cases  and  Amount  of  Difference.* 


A 

A 

A 

A 

A 

$r. 

pTe! 

$r. 

pTe 

$  r. 

pTe 

%  >'■ 

P.  E. 

<f.r. 

P.  E. 

50 

.00 

60 

.38 

70 

.78 

80 

1.25 

90 

1.90 

51 

.04 

61 

.41 

71 

.82 

81 

1.30 

91 

1.99 

52 

.07 

62 

.45 

72 

.86 

82 

1.36 

92 

2.08 

53 

.11 

63 

.49 

73 

.91 

83 

1.41 

93 

2.19 

54 

.15 

64 

.53 

74 

.95 

84 

1.47 

94 

2.31 

55 

.19 

65 

.57 

75 

1.00 

85 

1.54 

95 

2.44 

56 

.22 

66 

.61 

76 

1.05 

86 

1.60 

96 

2.60 

57 

.26 

67 

.65 

77 

1.10 

87 

1.67 

97 

2.79 

58 

.30 

68 

.69 

78 

1.14 

88 

1.74 

98 

3.05 

59 

.34 

69 

.74 

79 

1.20 

89 

1.82 

99 

3.45 

*From  page  16  of  'The  Perception  of  Small  Differences, '   by  Fullerton  and 
Cattell. 


REFERENCES  FOR  FURTHER  STUDY.  165 

The  table  for  estimating  the  P.  E.  from  the  percentage  of  right 
cases  for  a  given  difference  is  so  frequently  useful  that  I  reprint  it 
here  for  the  sake  of  those  to  whom  the  monograph  may  be  inaccessible. 
'  Hereditary  Genius.'     By  Francis  Galton.     Chapters  1,  2  and  3. 
'  The  Correlation  of  Mental  and  Physical  Measurements.'     By  Clark 

Wissler.     Monograph  Supplement,  Xo.  16,  to  the  Psychological 

Review. 
'  Natural  Inheritance.'     By  Francis  Galton.     Chapters  8  and  9. 
'  Statistics  of  American   Psychologists.'      By  J.    McKeen  Cattell. 

American  Journal  of  Psychology,  Vol.  XIV.,  pp.  310-328. 

The  last  two  studies  illustrate  the  importance  of  measures  by  rela- 
tive position.  Since  such  measures  are  likely  to  be  of  great  service 
in  the  social  sciences  and  in  scientific  studies  of  history  and  litera- 
ture, these  articles  may  well  be  examined  by  other  than  psychological 
students. 

Education. 

*  The  Age  of  Graduation  from  College.'     By  Win  field  Scott  Thomas. 
Popular  Science  Monthly,  June,  1903. 

The  article  by  Thomas,  though  extremely  simple,  is  a  most  use- 
ful illustration  of  the  value  of  other  measures  than  the  average  for  a 
central  tendency  and  of  the  significance  of  measures  of  variability. 
'  The  Correlations  of  the  Abilities  Involved  in  Secondary  School 
Work.'     By  W.  P.  Burris.     In  Heredity,  Correlation  and  Sex 
Differences  in  School  Abilities;   Columbia  Contributions  to  Phi- 
losophy, Psychology  and  Education,  Vol.  XI.,  No.  2. 
This  article  represents  a  condensed  report.     Hence  the  method 
used  is  incompletely  described  and  the  original  data  are  omitted.     The 
article  is,  however,  valuable  as  a  suggestion  of  the  susceptibility  of 
even  complex  educational  problems  to  exact  quantitative  study.     In 
spite  of  the  wealth  of  material  at  hand    in  school  reports,  teacher's 
records  and  the  like,  the  author  can  find  no  better  samples  of  the  use 
of  modern  statistical  methods  in  educational   science  than  these  two 
slight  studies. 

Economics  and  Social  Science. 

( Elements  of  Statistics.'     l!y  A.  L.  Bowley. 

This  book,  besides  giving  a  general  account  of  statistical  procedure 
in  economics,  contains  many  samples  of  facts  and  relations  adequately 


166  MENTAL   AND  SOCIAL  MEASUREMENTS. 

described  and  a  comparatively  simple  account  of  the  application  of 

the  theory  of  probability  to  measurements  of  facts. 

•  Notes  on  the  History  of  Pauperism  in  England  and  "Wales  from 
L850,  treated  by  the  method  of  frequency-curves  ;  with  an  intro- 
duction on  the  method.'  By  G.  Udney  Yule,  Journal  of  the  Royal 
Statistical  Society,  June,  1896. 

'  On  the  Correlation  of  Total  Pauperism  with  Proportion  of  Outdoor 
Relief.'  By  G.  Udney  Yule.  Economic  Journal,  December, 
1895,  and  December,  1896. 

4  An  Investigation  into  the  Causes  of  Changes  in  Pauperism,  in  Eng- 
land Chiefly  during  the  last  Two  Intercensal  Periods.'  By  G. 
Udney  Yule.    Journal  of  the  Royal  Statistical  Society,  June,  1899. 

Professor  Yule's  articles  on  pauperism  represent  the  application 
of  modern  methods  of  measurement  to  the  economic  and  social  sci- 
ences. They  illustrate  the  advantages  to  be  gained  in  these  sciences 
from  dealing  with  total  distributions  rather  than  averages  and  from 
using  appropriate  methods  of  measuring  variable  relationships.  By 
means  of  Pearson  coefficients  of  correlation,  Professor  Yule  was  able 
to  turn  certain  data  on  pauperism  to  a  new  use.  Care  in  the  mathe- 
matical handling  of  the  measures  used  is  also  well  shown.  In  respect 
to  wise  choice  of  units  and  a  vivid  sense  of  the  concrete  facts  repre- 
sented by  the  measures,  the  articles  are  more  questionable. 

Anthropometry. 

'  Natural  Inheritance.'     By  Francis  Galton.     Chapters  1—7. 

'  The  Growth  of  United  States  Naval  Cadets.  '     By  H.  G.  Beyer. 

Proceedings  of  the  United  States  Naval  Institute.     Vol.  21  (1895), 

pp.  297-333. 

The  present  activity  on  the  part  of  English  men  of  science  in 
developing  methods  of  exact  measurement  of  variable  phenomena 
had  its  source  in  Galton's  work.  This  book  is  therefore  a  fitting 
introduction  for  the  student  because  of  its  historical  importance  as 
well  as  the  relative  simplicity  of  its  mathematics.  Dr.  Beyer's  arti- 
cle is  still  simpler  in  its  manner  of  presentation,  but  is  unfortunately 
inaccessible  to  most  students. 
'  The  Growth  of  Boys.'     By  C.  "Wissler.     American  Anthropologist, 

New  Series,  Vol.  V.,  No.  1. 


REFERENCES  FOR  FURTHER  STUDY.  167 

The  article  by  Wissler  reports  one  of  the  very  few  studies  of. 
change  in  which  changes  themselves  are  measured.  It  demonstrates 
in  a  most  elegant  manner  the  law  of  compensation  by  which  rela- 
tively slow  growth  up  to  a  certain  age  implies  relatively  rapid  growth 
thereafter.  If  the  material  had  been  lumped  into  undistributed 
averages  in  the  customary  way  none  of  the  author's  conclusions  could 
have  been  reached. 
'  The  Cephalic  Index.'     By  Franz  Boas.     American  Anthropologist, 

New  Series,  Vol.  I.,  pp.  448-461. 
'  The  Growth  of  Toronto  Children.'     By  Franz  Boas.     Report  of 

the  United  States  Commissioner  of  Education  for  1896-97,  Vol. 

2,  pp.  1541-1599. 
'  On   the  Variability  and  Correlation   of  the   Hand.'     By   M.   A. 

AVhiteley  and  Karl  Pearson.     Proceedings  of  the  Royal  Society 

of  London,  Vol.  65,  pp.  126—151. 
'  On  the   Variability  and  Correlation  of  the  Hand.'     By  M.  A. 

Lewenz  and  M.  A.  Whiteley.     Biometrika,  Vol.  I. 

The  first  article  by  Boas  is  especially  interesting  as  an  illustra- 
tion of  the  uses  of  exact  statistical  methods  in  elucidating  causes. 
The  second  article  by  Boas  and  the  articles  on  the  anatomy  of  the 
hand,  report  studies  made  with  extreme  quantitative  refinement  and 
presented  in  full  detail. 

Vital  Statistics. 

'The  Chances  of  Death.'     By  Karl  Pearson.     In  a  volume  with  the 

same  title. 
( Zur   Theorie     der    Massenerscheinungen    in     der    Menschlichen 

Gesellschaft.'     By  W.  Lexis. 
'  On  the  Inheritance  of  the  Duration  of  Life.'     By  Mary  Beeton  and 

Karl  Pearson.     Biometrika,  Vol.  I. 

Biology. 
1  Statistical  Methods.'     By  C.  B.  Davenport. 

1  Die    Methode   der   Variations-Statistik.'     G.  Duncker.     Arch.  f. 
Entwickelungs- Median,  d.  Organismm,  VIII.,  112-183. 
For  further  references  see  the  bibliographies  given  by  Davenport 
and  Duncker. 

Pure  Statistics. 
'The  Principles  of  Science.'      W.  S.  Jevons. 
'  The  Logic  of  Chance.'     J.  Venn. 


L68  MENTAL  AND  SOCIAL  MEASUREMENTS. 

The  chapters  od  permutations,  combinations  and  probability  in  any 

standard  algebra. 

'  History  of  the  Theory  of  Probability.'     I.  Todhunter. 

'The  Method  of  Least  Squares.'     M.  Merriman. 

'  Hereditary  Genius.'     F.  Galton.     Chapters  1-3. 

'  Natural  Inheritance.'     F.  Galton.     Chapters  1-7. 

'  Lettres  sur  la  Probability.'     A.  Quetelet.     (Difficult  of  access.) 

'Elements  of  Statistics.'     A.  L.  Bowley.     Part  II. 

'  Grammar  of  Science '  (second  edition).     Karl  Pearson.     Chapters 

X.-XI. 
'  Theorie  der  Bevolkerungs  und  Moralstatistik.'    W.  Lexis.    Chap- 
ter VI. 
'  On  the  Theory  of  Correlation.'     G.  U.  Yule.     Journal  of  the  Royal 

Statistical  Society,  Vol.  60,  pp.  812-854. 
1  Collektivmasslehre.'     G.  T.  Fechner. 
'  The  Proof  and  Measurement  of  Association  Between  Two  Things.' 

C.  Spearman.     American  Journal  of  Psychology,  January,  1904, 

Vol.  XV.,  pp.  72-101. 

Material  for  more  advanced  study  of  pure  statistics  will  be  found 
in  the  writings  of  Franz  Boas,  H.  Bruns,  F.  Y.  Edgeworth,  W. 
Lexis,  G.  Lipps,  Karl  Pearsoti,  "W.  F.  Sheppard,  H.  Westergaard 
and  G.  U.  Yule. 

The  contributions  of  the  English  students  of  pure  statistics  will 
be  found  chiefly  in  the  Philosophical  Transactions  of  the  Poyal  Society 
of  London,  in  the  Proceedings  of  the  same  society,  in  the  Journal  of  the 
Boyal  Statistical  Society,  in  Biometrika,  and  in  the  London,  Edin- 
burgh, and  Dublin  Philosophical  Magazine  and  Journal  of  Science. 

Special  lists  of  references  to  both  pure  and  applied  statistics  will 
be  found  in  Bowley's  '  Elements  of  Statistics,'  Davenport's  '  Statis- 
tical Methods  '  and  Duncker's  '  Methode  der  Variations-Statistik.' 


APPENDIX   I. 

A   MULTIPLICATION   TABLE    UP   TO    100  X  100. 

The  reader's  attention  has  already  been  called  to  Crelle's  Rech- 
entafeln,  a  multiplication  table  up  to  1000  x  1000.  It  saves  much 
time,  replaces  mental  work  by  finger  and  eye  work,  and  decreases 
errors  in  calculation.  Crelle's  table,  however,  makes  a  book  some  9 
by  14  inches,  weighing  several  pounds.  The  table  that  follows  is  a 
modification  of  Crelle's  table,  but  runs  only  to  100  x  100.  For 
work  with  these  smaller  numbers  and  for  approximate  calculations, 
it  is  more  rapid  than  the  longer  table  and  is  so  arranged  as  to  be 
easier  for  the  eyes. 

Its  uses  will  be  apparent  upon  examination,  but  the  reader  should 
note  that  it  serves  for  division  as  well  as  for  multiplication.  In 
dividing,  one  of  course  finds  the  divisor  in  the  row  of  figures  in  heavy 
faced  type  at  the  top  of  the  page,  hunts  for  the  dividend  in  the  col- 
umn beneath  it,  and,  this  being  found,  obtains  the  quotient  in  the 
figure  in  heavy-faced  type  at  the  side  of  the  page.  Thus  to  divide 
684  by  38,  one  looks  under  38,  finds  684  and  opposite  it,  at  the  side 
of  the  page,  18,  the  answer.  Again  to  divide  1,600  by  38,  one  looks 
under  38,  finds  1596  to  be  the  nearest  number,  and  so  the  nearest 
two-figure  answer  to  be  42.  If  one  needed  greater  precision,  he  could 
divide  the  remainder  4.0  by  38,  getting  0.1,  and  then  the  remainder 
.2000,  getting  .0052,  or  42.1052,  and  so  on  to  any  desired  precision. 


1G9 


170  MENTAL    AND  SOCIAL  MEASUREMENTS. 

23456789        10 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

2 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

3 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

4 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

5 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

6 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

7 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

8 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

9 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

10 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

11 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

12 

13 

26 

39 

52 

65 

78 

91 

104 

117 

130 

13 

14 

28 

42 

56 

70 

84 

98 

112 

126 

140 

14 

15 

30 

45 

60 

75 

90 

105 

120 

135 

150 

15 

16 

32 

48 

64 

80 

96 

112 

128 

144 

160 

16 

17 

34 

51 

68 

85 

102 

119 

136 

153 

170 

17 

18 

36 

54 

72 

90 

108 

126 

144 

162 

180 

18 

19 

38 

57 

76 

95 

114 

133 

152 

171 

190 

19 

20 

40 

60 

80 

100 

120 

140 

160 

180 

200 

20 

21 

42 

63 

84 

105 

126 

147 

168 

189 

210 

21 

22  - 

44 

66 

88 

110 

132 

154 

176 

198 

220 

22 

23 

46 

69 

92 

115 

138 

161 

184 

207 

230 

23 

24 

48 

72 

96 

120 

144 

168 

192 

216 

240 

24 

25 

50 

75 

100 

125 

150 

175 

200 

225 

250 

25 

26 

52 

78 

104 

130 

156 

182 

208 

234 

260 

26 

27 

54 

81 

108 

135 

162 

189 

216 

243 

270 

27 

28 

56 

84 

112 

140 

168 

196 

224 

252 

280 

28 

29 

58 

87 

116 

145 

174 

203 

232 

261 

290 

29 

30 

60 

90 

120 

150 

180 

210 

240 

270 

300 

30 

31 

62 

93 

124 

155 

186 

217 

248 

279 

310 

31 

32 

64 

96 

128 

160 

192 

224 

256 

288 

320 

32 

33 

66 

99 

132 

165 

198 

231 

264 

297 

330 

33 

34 

68 

102 

136 

170 

204 

238 

272 

306 

340 

34 

35 

70 

105 

140 

175 

210 

245 

280 

315 

350 

35 

36 

72 

108 

144 

180 

216 

252 

288 

324 

360 

36 

37 

74 

111 

148 

185 

222 

259 

296 

333 

370 

37 

38 

76 

114 

152 

190 

228 

266 

304 

342 

380 

38 

39 

78 

117 

156 

195 

234 

273 

312 

351 

390 

39 

40 

80 

120 

160 

200 

240 

280 

320 

360 

400 

40 

41 

82 

123 

164 

205 

246 

287 

328 

369 

410 

41 

42 

84 

126 

168 

210 

252 

294 

336 

378 

420 

42 

43 

86 

129 

172 

215 

258 

301 

344 

'387 

430 

43 

44 

88 

132 

176 

220 

264 

308 

352 

396 

440 

44 

45 

90 

135 

180 

225 

270 

315 

360 

405 

450 

45 

46 

92 

138 

184 

230 

276 

322 

368 

414 

460 

46 

47 

94 

141 

188 

235 

282 

319 

376 

423 

470 

47 

48 

96 

144 

192 

240 

288 

336 

384 

432 

480 

48 

49 

98 

147 

196 

245 

294 

343 

392 

441 

490 

49 

50 

100 

150 

200 

250 

300 

350 

400 

450 

500 

50 

10 


A   MULTIPLICATION  TABLE.  171 

4  5  6  7         8         9        10 


51 

102 

153 

204 

255 

306 

357 

408 

459 

510 

51 

52 

104 

156 

208 

260 

312 

364 

416 

468 

520 

52 

53 

106 

159 

212 

265 

318 

371 

424 

477 

530 

53 

54 

108 

162 

216 

270 

324 

378 

432 

486 

540 

54 

55 

110 

165 

220 

275 

330 

385 

440 

495 

550 

55 

56 

112 

168 

224 

280 

336 

392 

448 

504 

560 

56 

57 

114 

171 

228 

285 

342 

399 

456 

513 

570 

57 

58 

116 

174 

232 

290 

348 

406 

464 

522 

580 

58 

59 

118 

177 

236 

295 

354 

413 

472 

531 

590 

59 

60 

120 

180 

240 

300 

360 

420 

480 

540 

600 

60 

61 

122 

183 

244 

305 

366 

427 

488 

549 

610 

61 

62 

124 

186 

248 

310 

372 

434 

496 

558 

620 

62 

63 

126 

189 

252 

315 

378 

441 

504 

567 

630 

63 

64 

128 

192 

256 

320 

384 

448 

512 

576 

640 

64 

65 

130 

195 

260 

325 

390 

455 

520 

585 

650 

65 

66 

132 

198 

264 

330 

396 

462 

528 

594 

660 

66 

67 

134 

201 

268 

335 

402 

469 

536 

603 

670 

67 

68 

136 

204 

272 

340 

408 

476 

544 

612 

680 

68 

69 

138 

207 

276 

345 

414 

483 

552 

621 

690 

69 

70 

140 

210 

280 

350 

420 

490 

560 

630 

700 

70 

71 

142 

213 

284 

355 

426 

497 

568 

639 

710 

71 

72 

144 

216 

288 

360 

432 

504 

576 

648 

720 

72 

73 

146 

219 

292 

365 

438 

511 

584 

657 

730 

73 

74 

148 

222 

296 

370 

444 

518 

592 

666 

740 

74 

75 

150 

225 

300 

375 

450 

525 

600 

675 

750 

75 

76 

152 

228 

304 

380 

456 

532 

608 

684 

760 

76 

77 

154 

231 

308 

385 

462 

539 

616 

693 

770 

77 

78 

156 

234 

312 

390 

468 

546 

624 

702 

780 

78 

79 

158 

237 

316 

395 

474 

553 

632 

711 

790 

79 

80 

160 

240 

320 

400 

480 

560 

640 

720 

800 

80 

81 

162 

243 

324 

405 

486 

567 

648 

729 

810 

81 

82 

164 

246 

328 

410 

492 

574 

656 

738 

820 

82 

83 

166 

249 

332 

415 

498 

581 

664 

747 

830 

83 

84 

168 

252 

336 

420 

504 

588 

672 

756 

840 

84 

85 

170 

255 

340 

425 

510 

595 

680 

765 

850 

85 

86 

172 

258 

344 

430 

516 

602 

688 

774 

860 

86 

87 

174 

261 

348 

435 

522 

609 

696 

783 

870 

87 

88 

176 

264 

352 

440 

528 

616 

704 

792 

880 

88 

89 

178 

267 

356 

445 

534 

623 

712 

801 

890 

89 

90 

180 

270 

360 

450 

540 

630 

720 

810 

900 

90 

91 

182 

273 

364 

455 

546 

637 

728 

819 

910 

91 

92 

184 

276 

3CS 

460 

552 

644 

736 

828 

920 

92 

93 

L86 

279 

372 

465 

558 

651 

744 

837 

930 

93 

94 

188 

282 

376 

470 

564 

658 

752 

846 

940 

94 

95 

190 

285 

380 

475 

570 

o\:, 

760 

855 

950 

95 

96 

192 

288 

384 

480 

576 

672 

768 

-SCI 

960 

96 

97 

194 

291 

388 

485 

582 

679 

776 

ST.". 

'.(71 ) 

97 

98 

196 

294 

392 

490 

588 

686 

784 

882 

980 

98 

99 

L98 

297 

396 

495 

594 

693 

792 

891 

990 

99 

100 

200 

300 

400 

500 

COO 

700 

800 

900 

1000 

100 

10 


172  MENTAL  AND  SOCIAL  MEASUREMENTS. 

11        12        13        14        15        16        17        18        19       20 


1 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

1 

2 

oo 

24 

26 

28 

30 

32 

34 

36 

38 

40 

2 

3 

33 

36 

39 

42 

45 

48 

51 

54 

57 

60 

3 

4 

44 

48 

52 

56 

60 

64 

68 

72 

76 

80 

4 

5 

55 

60 

65 

70 

75 

80 

85 

90 

95 

100 

5 

6  66  72  78  84  90  96  102  108  114  120  6 

7  77  84  91  98  105  112  119  126  133  140  7 

8  88  96  104  112  120  128  136  144  152  160  8 

9  99  108  117  126  135  144  153  162  171  *  180  9 
10  110  120  130  140  150  160  170  180  190  200  10 


11 

121 

132 

143 

154 

165 

176 

187 

198 

209 

220 

11 

12 

132 

144 

156 

168 

180 

192 

204 

216 

228 

240 

12 

13 

143 

156 

169 

182 

195 

208 

221 

234 

247 

260 

13 

14 

154 

168 

182 

196 

210 

224 

238 

252 

266 

280 

14 

15 

165 

180 

195 

210 

225 

240 

255 

270 

285 

300 

15 

16 

176 

192 

208 

224 

240 

256 

272 

288 

304 

320 

16 

17 

187 

204 

221 

238 

255 

272 

289 

306 

323 

340 

17 

18 

198 

216 

234 

252 

270 

288 

306 

324 

342 

360 

18 

19 

209 

228 

247 

266 

285 

304 

323 

342 

361 

380 

19 

20 

220 

240 

260 

280 

300 

320 

340 

360 

380 

400 

20 

21 

231 

252 

273 

294 

315 

336 

357 

378 

399 

420 

21 

22 

242 

264 

286 

308 

330 

352 

374 

396 

418 

440 

22 

23 

253 

276 

299 

322 

345 

368 

391 

414 

437 

460 

23 

24 

264 

28S 

312 

336 

360 

384 

408 

432 

456 

480 

24 

25 

275 

300 

325 

350 

375 

400 

425 

450 

475 

500 

25 

26 

286 

312 

338 

364 

390 

416 

442 

468 

494 

520 

26 

27 

297 

324 

351 

378 

405 

432 

459 

486 

513 

540 

27 

28 

308 

336 

364 

392 

420 

448 

476 

504 

532 

560 

28 

29 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

29 

30 

330 

360 

390 

420 

450 

480 

510 

540 

570 

600 

30 

31 

341 

372 

403 

434 

465 

496 

527 

558 

589 

620 

31 

32 

352 

384 

416 

448 

480 

512 

544 

576 

608 

640 

32 

33 

363 

396 

429 

462 

495 

528 

561 

594 

627 

660 

33 

34 

374 

408 

442 

476 

510 

544 

578 

612 

646 

680 

34 

35 

385 

420 

455 

490 

525 

560 

595 

630 

665 

700 

35 

36 

396 

432 

468 

504 

540 

576 

612 

648 

684 

720 

36 

37 

407 

444 

481 

518 

555 

592 

629 

666 

703 

740 

37 

38 

418 

456 

494 

532 

570 

608 

646 

684 

722 

760 

38 

39 

429 

468 

507 

546 

585 

624 

663 

702 

741 

780 

39 

40  440       480       520       560       600       640       680       720       760       800  40 

41  451       492       533       574       615       656       697       738       779       820  41 


42 

462 

504 

546 

588 

630 

672 

714 

756 

798 

840 

42 

43 

473 

516 

559 

602 

645 

688 

731 

774 

817 

860 

43 

44 

484 

528 

572 

616 

660 

704 

748 

792 

836 

880 

44 

45 

495 

540 

585 

630 

675 

720 

765 

810 

855 

900 

45 

46 

506 

552 

598 

644 

690 

736 

782 

828 

874 

920 

46 

47 

517 

564 

611 

658 

705 

752 

799 

846 

893 

940 

47 

48 

528 

576 

624 

672 

720 

768 

816 

864 

912 

960 

48 

49 

539 

588 

637 

686 

735 

784 

833 

882 

931 

980 

49 

50 

550 
11 

600 
12 

650 
13 

700 
14 

750 
15 

800 
16 

850 
17 

900 
18 

950 
19 

1000 
20 

50 

A  MULTIPLICATION  TABLE.  173 

11        12        13       14        15       16        17       18        19       20 


51 

561 

612 

663 

714 

765 

816 

867 

918 

969 

1020 

51 

52 

572 

624 

676 

728 

780 

832 

884 

936 

988 

1040 

52 

53 

583 

636 

689 

742 

795 

848 

901 

954 

1007 

1060 

53 

54 

594 

648 

702 

756 

810 

864 

918 

972 

1026 

1080 

54 

55 

605 

660 

715 

770 

S25 

880 

935 

990 

1045 

1100 

55 

56 

616 

672 

728 

784 

840 

896 

952 

1008 

1064 

1120 

56 

57 

627 

684 

741 

798 

855 

912 

969 

1026 

1083 

1140 

57 

58 

638 

696 

754 

812 

870 

928 

986 

1044 

1102 

1160 

58 

59 

649 

708 

767 

826 

885 

944 

1003 

1002 

1121 

1180 

59 

60 

660 

720 

780 

840 

900 

960 

1020 

1080 

1140 

1200 

60 

61 

671 

732 

793 

854 

915 

976 

1037 

1098 

1159 

1220 

61 

62 

682 

744 

806 

868 

930 

992 

1054 

1116 

1178 

1240 

62 

63 

693 

756 

819 

882 

945 

1008 

1071 

1134 

1197 

1260 

63 

64 

704 

768 

832 

896 

960 

1024 

1088 

1152 

1216 

1280 

64 

65 

715 

780 

845 

910 

975 

1040 

1105 

1170 

1235 

1300 

65 

66 

726 

792 

858 

924 

990 

1056 

1122 

1188 

1254 

1320 

66 

67 

737 

804 

871 

938 

1005 

1072 

1139 

1206 

1273 

1340 

•67 

68 

748 

816 

884 

952 

1020 

1088 

1156 

1224 

1292 

1360 

68 

69 

759 

828 

897 

966 

1035 

1104 

1173 

1242 

1311 

1380 

69 

70 

770 

840 

910 

980 

1050 

1120 

1190 

1260 

1330 

1400 

70 

71 

781 

852 

923 

994 

1065 

1136 

1207 

1278 

1349 

1420 

71 

72 

792 

864 

936 

1008 

1080 

1152 

1224 

1296 

1368 

1440 

72 

73 

803 

876 

949 

1022 

1095 

1168 

1241 

1314 

1387 

1460 

73 

74 

814 

888 

962 

1036 

1110 

1184 

1258 

1332 

1406 

1480 

74 

75 

825 

900 

975 

1050 

1125 

1200 

1275 

1350 

1425 

1500 

75 

76 

836 

912 

988 

1064 

1140 

1216 

1292 

1368 

1444 

1520 

76 

77 

847 

924 

1001 

1078 

1155 

1232 

1309 

1386 

1463 

1540 

77 

78 

858 

936 

1014 

1092 

1170 

1248 

1326 

1404 

1482 

1560 

78 

79 

869 

948 

1027 

1106 

1185 

1204 

1343 

1422 

1501 

1580 

79 

80 

880 

960 

1040 

1120 

1200 

1280 

1360 

1440 

1520 

1600 

80 

81 

891 

972 

1053 

1134 

1215 

1296 

1377 

1458 

1539 

1620 

81 

82 

902 

984 

1066 

1148 

1230 

1312 

1394 

1476 

1558 

1640 

82 

83 

913 

996 

1079 

1162 

1245 

1328 

1411 

1494 

1577 

1660 

83 

84 

924 

1008 

1092 

1176 

1260 

1344 

1428 

1512 

1596 

1680 

84 

85 

935 

1020 

1105 

1190 

1275 

1360 

1445 

1530 

1615 

1700 

85 

86 

946 

1032 

1118 

1204 

1290 

1376 

1462 

1548 

1634 

1720 

86 

87 

957 

1044 

1131 

1218 

1305 

1392 

1479 

1566 

1653 

1740 

87 

88 

968 

1056 

1144 

1232 

1320 

1408 

1496 

1584 

1672 

1760 

88 

89 

979 

1068 

1157 

1246 

1335 

1424 

1513 

1602 

1691 

1780 

89 

90 

990 

1080 

1170 

1260 

1350 

1440 

1530 

1620 

1710 

1800 

90 

91 

1001 

1092 

1183 

1274 

1365 

1456 

1547 

1638 

1729 

1820 

91 

92 

1012 

1104 

1196 

1288 

1380 

1472 

L564 

L656 

1748 

IS  10 

92 

93 

1023 

L116 

1209 

L302 

1395 

1488 

l.-.xi 

107  1 

1707 

I860 

93 

94 

1034 

1128 

1 222 

1316 

1410 

1504 

1598 

1692 

17S0 

1SS0 

94 

95 

1045 

1140 

1235 

1330 

1425 

1520 

L615 

1710 

1X05 

1000 

95 

96 

lo:,  6 

1 1 52 

1248 

13  11 

1  IK) 

L536 

1632 

1728 

1824 

L920 

96 

97 

1067 

L164 

1261 

1358 

1455 

1552 

10  10 

1746 

1843 

1940 

97 

98 

1078 

1176 

1274 

L372 

1470 

1568 

1000 

1701 

L862 

1000 

98 

99 

1089 

1188 

1287 

L386 

1 485 

1 58  1 

1683 

L782 

1881 

10X0 

99 

100 

uoo 

1200 

1300 

1400 

1500 

1600 

1700 

1800 

1900 

2000 

100 

11        12        13        14        15       16        17        18        19        20 


21 

22 

23 

.24 

25 

26 

27 

28 

29 

30 

1 

■12 

44 

46 

48 

50 

52 

54 

56 

58 

60 

2 

63 

66 

69 

72 

75 

78 

81 

84 

87 

90 

3 

si 

88 

92 

96 

100 

104 

108 

112 

116 

120 

4 

105 

110 

115 

120 

125 

130 

135 

140 

145 

150 

5 

126 

132 

138 

144 

150 

156 

162 

168 

174 

180 

6 

147 

154 

161 

168 

175 

182 

189 

196 

203 

210 

7 

168 

176 

184 

192 

200 

208 

216 

224 

232 

240 

8 

189 

198 

207 

216 

225 

234 

243 

252 

261 

270 

9 

174  MENTAL   AND  SOCIAL  MEASUREMENTS. 

21   22   23   24   25   26   27   28   29   30 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10  210   220   230   240   250   260   270   280   290   300     10 

11  231   242   253   264   275   286   297   308   319   330     11 

12  252  264  276  288  300  312  324  336  348  360  12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


31 
32 
33 
34 
35 
36 
37 

38  798  836  874  912   950  988  1026  1064  1102  1140  38 

39  819  858  897  936   975  1014  1053  1092  1131  1170  39 

40  840  880  920  960  1000  1040  1080  1120  1160  1200  40 

41  861  902  943  984  1025  1066  1107  1148  1189  1230  42 

42  882  924  966  1008  1050  1092  1134  1176  1218  1260  41 

43  903  946  989  1032  1075  1118  1161  1204  1247  1290  43 

44  924  968  1012  1056  1100  1144  1188  1232  1276  1320  44 

45  945  990  1035  1080  1125  1170  1215  1260  1305  1350  45 

46  966  1012  1058  1104  1150  1196  1242  1288  1334  1380  46 

47  987  1034  1081  1128  1175  1222  1269  1316  1363  1410  47 

48  1008  1056  1104  1152  1200  1248  1296  1344  1392  1440  48 

49  1029  1078  1127  1176  1225  1274  1323  1372  1421  1470  49 

50  1050  1100  1150  1200  1250  1300  1350  1400  1450  1500  50 

21  22  23  24   25  26   27  28  29  30 


273 

286 

299 

312 

325 

338 

351 

364 

377 

390 

13 

294 

308 

322 

336 

350 

364 

378 

392 

406 

420 

14 

315 

330 

345 

360 

375 

390 

405 

420 

435 

450 

15 

336 

352 

368 

384 

400 

416 

432 

448 

464 

480 

16 

357 

374 

391 

408 

425 

442 

459 

476 

493 

510 

17 

378 

396 

414 

432 

450 

468 

486 

504 

522 

540 

18 

399 

418 

437 

456 

475 

494 

513 

532 

551 

570 

19 

420 

440 

460 

480 

500 

520 

540 

560 

580 

600 

20 

441 

462 

483 

504 

525 

546 

567 

588 

609 

630 

21 

462 

484 

506 

528 

550 

572 

594 

616 

638 

660 

22 

483 

506 

529 

552 

575 

598 

621 

644 

667 

690 

23 

504 

528 

552 

576 

600 

624 

648 

672 

696 

720 

24 

525 

550 

575 

600 

625 

650 

675 

700 

725 

750 

25 

546 

572 

598 

624 

650 

676 

702 

728 

754 

780 

26 

567 

594 

621 

648 

675 

702 

729 

756 

783 

810 

27 

588 

616 

644 

672 

700 

728 

756 

784 

812 

840 

28 

609 

638 

667 

696 

725 

754 

783 

812 

841 

870 

29 

630 

660 

690 

720 

750 

780 

810 

840 

870 

900 

30 

651 

682 

713 

744 

775 

806 

837 

868 

899 

930 

31 

672 

704 

736 

768 

800 

832 

864 

896 

928 

960 

32 

693 

726 

759 

792 

825 

858 

891 

924 

957 

990 

33 

714 

748 

782 

816 

850 

884 

918 

952 

986 

1020 

34 

735 

770 

805 

840 

875 

910 

945 

980 

1015 

1050 

35 

756 

792 

828 

864 

900 

936 

972 

1008 

1044 

1080 

36 

777 

814 

851 

888 

925 

962 

999 

1036 

1073 

1110 

37 

A  MULTIPLICATION  TABLE.  175 

21  22  23  24  25  26  27  28  29  30 

51  1071  1122  1173  1224  1275  1326  1377  •  1428  1479  1530  51 

52  1092  1144  1196  1248  1300  1352  1404  1456  1508  1560  52 

53  1113  1166  1219  1272  1325  1378  1431  1484  1537  1590  53 

54  1134  1188  1242  1296  1350  1404  1458  1512  1566  1620  54 

55  1155  1210  1265  1320  1375  1430  1485  1540  1595  1650  55 

56  1176  1232  1288  1344  1400  1456  1512  1568  1624  16S0  56 

57  1197  1254  1311  1368  1425  1482  1539  1596  1653  1710  57 

58  1218  1276  1334  1392  1450  1508  1566  1624  1682  1740  58 

59  1239  1298  1357  1416  1475  1534  1593  1652  1711  1770  59 

60  1260  1320  1380  1440  1500  1560  1620  1680  1740  1800  60 

61  1281  1342  1403  1464  1525  1586  1647  1708  1769  1830  61 

62  1302  1364  1426  1488  1550  1612  1674  1736  1798  1860  62 

63  1323  1386  1449  1512  1575  1638  1701  1764  1827  1890  63 

64  1344  1408  1472  1536  1600  1664  1728  1792  1856  1920  64 

65  1365  1430  1495  1560  1625  1690  1755  1820  1885  1950  65 

66  1386  1452  1518  1584  1650  1716  1782  1848  1914  1980  66 

67  1407  1474  1541  1608  1675  1742  1809  1876  1943  2010  67 

68  1428  1496  1564  1632  1700  1768  1836  1904  1972  2040  68 

69  1449  1518  1587  1656  1725  1794  1863  1932  2001  2070  69 

70  1470  1540  1610  1680  1750  1820  1890  1960  2030  2100  70 

71  1491  1562  1633  1704  1775  1846  1917  1988  2059  2130  71 

72  1512  1584  1656  1728  1800  1872  1944  2016  2088  2160  72 

73  1533  1606  1679  1752  1825  1898  1971  2044  2117  2190  73 

74  1554  1628  1702  1776  1850  1924  1998  2072  2146  2220  74 

75  1575  1650  1725  1800  1875  1950  2025  2100  2175  2250  75 

76  1596  1672  1748  1824  1900  1976  2052  2128  2204  2280  76 

77  1617  1694  1771  1848  1925  2002  2079  2156  2233  2310  77 

78  1638  1716  1794  1872  1950  2028  2106  2184  2262  2340  78 

79  1659  1738  1817  1896  1975  2054  2133  2212  2291  2370  79 

80  1680  1760  1840  1920  2000  2080  2160  2240  2320  2400  80 

81  1701  1782  1863  1944  2025  2106  2187  2268  2349  2430  81 
-82  1722  1804  188,6  1968  2050  2132  2214  2296  2378  2460  82 

83  1743  1826  1909  1992  2075  2158  2241  2324  2407  2490  83 

84  1764  1848  1932  2016  2100  2184  2268  2352  2436  2520  84 

85  1785  1870  1955  2040  2125  2210  2295  2380  2465  2550  85 

86  1806  1892  1978  2064  2150  2236  2322  2408  2494  2580  86 

87  1827  1914  2001  2088  2175  2262  2349  2436  2523  2610  87 

88  1848  1936  2021  2112  2200  2288  2376  2464  2552  2640  88 

89  1869  1958  2047  2136  2225  2314  2403  2492  2581  2670  89 

90  1890  1980  2070  2160  2250  2340  2430  2520  2610  2700  90 

91  1911  2002  2093  2184  2275  2366  2457  2548  2639  2730  91 

92  1932  2024  2116  2208  2300  2392  2484  2576  2668  2760  92 

93  1953  2046  2139  2232  2325  2418  2511  2604  2697  2790  93 

94  1974  2068  2162  2256  2350  2444  2538  2632  2726  2820  94 

95  1995  2090  2185  2280  2375  2470  2565  2660  2755  2850  95 

96  2016  2112  2208  2304  2400  2496  2592  2688  2784  2880  96 

97  2037  2134  2231  2328  2425  2522  2619  27  If!  2813  2910  97 

98  2058  2156  2254  2352  2450  2548  2646  2744  2842  29  10  98 

99  2079  2178  2277  2376  2475  2574  2673  2772  287]  2970  99 
100  2100  2200  2300  2400  2500  2600  2700  2800  2900  3000  100 

21  22  23  24  25  26  27  28  29  30 


176  MENTAL   AND  SOCIAL  MEASUREMENTS. 

31   32   33   34   35   36   37   38   39   40 


1 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

1 

2 

62 

64 

66 

68 

70 

72 

74 

76 

78 

80 

2 

3 

93 

96 

99 

K>2 

105 

108 

111 

114 

117 

120 

3 

4 

124 

1 28 

132 

136 

140 

144 

148 

152 

156 

160 

4 

5 

L55 

160 

L65 

170 

175 

180 

185 

190 

195 

200 

5 

6 

186 

192 

198 

204 

210 

216 

222 

228 

234 

240 

6 

7 

217 

221 

231 

238 

245 

252 

259 

266 

273 

280 

7 

8 

248 

256 

264 

272 

280 

288 

296 

304 

312 

320 

8 

9 

279 

288 

2H7 

306 

315 

324 

333 

342 

351 

360 

9 

10 

310 

320 

330 

340 

350 

360 

370 

380 

390 

400 

10 

11 

341 

352 

363 

374 

385 

396 

407 

418 

429 

440 

11 

12 

372 

384 

396 

408 

420 

432 

444 

456 

468 

480 

12 

13 

403 

416 

429 

442 

455 

468 

481 

494 

507 

520 

13 

14 

434 

448 

462 

476 

490 

504 

518 

532 

546 

560 

14 

15 

465 

480 

495 

510 

525 

540 

555 

570 

585 

600 

15 

16 

496 

512 

528 

544 

560 

576 

592 

608 

624 

640 

16 

17 

-327 

544 

561 

578 

595 

612 

629 

646 

663 

680 

17 

18 

558 

576 

594 

612 

630 

648 

666 

684 

702 

720 

18 

19 

589 

608 

627 

646 

665 

684 

703 

722 

741 

760 

19 

20 

620 

640 

660 

680 

700 

720 

740 

760 

780 

800 

20 

21 

651 

672 

693 

714 

735 

756 

777 

798 

819 

840 

21 

22 

682 

704 

726 

748 

770 

792 

814 

836 

858 

880 

22 

23 

713 

736 

759 

782 

805 

828 

851 

874 

897 

920 

23 

24 

744 

768 

792 

816 

840 

S64 

888 

912 

936 

960 

24 

25 

775 

800 

825 

850 

875 

900 

925 

950 

975 

1000 

25 

26 

806 

832 

858 

884 

910 

936 

962 

988 

1014 

1040 

26 

27 

837 

864 

891 

918 

945 

972 

999 

1026 

1053 

1080 

27 

28 

868 

896 

924 

952 

980 

1008 

1036 

1064 

1092 

1120 

28 

29 

899 

928 

957 

986 

1015 

1044 

1073 

1102 

1131 

1160 

29 

30 

930 

960 

990 

1020 

1050 

1080 

1110 

1140 

1170 

1200 

30 

31 

961 

992 

1023 

1054 

1085 

1116 

1147 

1178 

1209 

1240 

31 

32 

992 

1024 

1056 

1088 

1120 

1152 

1184 

1216 

1248 

1280 

32 

33 

1023 

1056 

1089 

1122 

1155 

1188 

1221 

1*254 

1287 

1320 

33 

34 

1054 

1088 

1122 

1156 

1190 

1224 

1258 

1292 

1326 

1360 

34 

35 

1085 

1120 

1155 

1190 

1225 

1260 

1295 

1330 

1365 

1400 

35 

36 

1116 

1152 

1188 

1224 

1260 

1296 

1332 

1368 

1404 

1440 

36 

37 

1147 

1184 

1221 

1258 

1295 

1332 

1369 

1406 

1443 

1480 

37 

38 

1178 

1216 

1254 

1292 

1330 

1368 

1406 

1444 

1482 

1520 

38 

39 

1209 

1248 

1287 

1326 

1365 

1404 

1443 

1482 

1521 

1560 

39 

40 

1240 

1280 

1320 

1360 

1400 

1440 

1480 

1520 

1560 

1600 

40 

41 

1271 

1312 

1353 

1394 

1435 

1476 

1517 

1558 

1599 

1640 

41 

42 

1302 

1344 

1386 

1428 

1470 

1512 

1554 

1596 

1638 

1680 

42 

43 

1333 

1376 

1419 

1462 

1505 

1548 

1591 

1634 

1677 

1720 

43 

44 

1364 

1408 

1452 

1496 

1540 

1584 

1628 

1672 

1716 

1760 

44 

45 

1395 

1440 

1485 

1530 

1575 

1620 

1665 

1710 

1755 

1800 

45 

46 

1426 

1472 

1518 

1564 

1610 

1656 

1702 

1748 

1794 

1840 

46 

47 

1457 

1504 

1551 

1598 

1645 

1692 

1739 

1786 

1833 

1880 

47 

48 

1488 

1536 

1584 

1632 

1680 

1728 

1776 

1824 

1872 

1920 

48 

49 

1519 

1568 

1617 

1666 

1715 

1764 

1813 

1862 

1911 

1960 

49 

50 

1550 

1600 

1650 

1700 

1750 

1800 

1850 

1900 

1950 

2000 

50 

31   32   33   34   35   36   37   38   39   40 


A  MULTIPLICATION  TABLE.  177 

31  32   33  34  35  36  37  38  39  40 

51  1581  1632  1683  1734  1785  1836  1887  1938  1989  2040  51 

52  1612  1664  1716  1768  1820  1872  1924  1976  2028  2080  52 

53  1643  1696  1749  1802  1855  1908  1961  2014  2067  2120  53 

54  1674  1728  1782  1836  1890  1944  1998  2052  2106  2160  54 

55  1705  1760  1815  1870  1925  1980  ■  2035  2090  2145  2200  55 

56  1736  1792  1848  1904  1960  2016  2072  2128  2184  2240  56 

57  1767  1824  1881  1938  1995  2052  2109  2166  2223  2280  57 

58  1798  1856  1914  1972  2030  2088  2146  2204  2262  2320  58 

59  1829  1888  1947  2006  2065  2124  2183  2242  2301  2360  59 

60  1860  1920  1980  2040  2100  2160  2220  2280  2340  2400  60 

61  1891  1952  2013  2074  2135  2196  2257  2318  2379  2440  61 

62  1922  1984  2046  2108  2170  2232  2294  2356  2418  2480  62 

63  1953  2016  2079  2142  2205  2268  2331  2394  2457  2520  63 

64  1984  2048  2112  2176  2240  2304  2368  2432  2496  2560  64 

65  2015  2080  2145  2210  2275  2340  2405  2470  2535  2600  65 

66  2046  2112  2178  2244  2310  2376  2442  2508  2574  2640  66 

67  2077  2144  2211  2278  2345  2412  2479  2546  2613  2680  67 

68  2108  2176  2244  2312  2380  2448  2516  2584  2652  2720  68 

69  2139  2208  2277  2346  2415  2484  2553  2622  2691  2760  69 

70  2170  2240  2310  2380  2450  2520  2590  2660  2730  2800  70 

71  2201  2272  2343  2414  2485  2556  2627  2698  2769  2840  71 

72  2232  2304  2376  2448  2520  2592  2664  2736  2808  2880  72 

73  2263  2336  2409  2482  2555  2628  2701  2774  2847  2920  73 

74  2294  2368  2442  2516  2590  2664  2738  2812  2886  2960  74 

75  2325  2400  2475  2550  2625  2700  2775  2850  2925  3000  75 

76  2356  2432  2508  2584  2660  2736  2812  2888  2964  3040  76 

77  2387  2464  2541  2618  2695  2772  2849  2926  3003  3080  77 

78  2418  2496  2574  2652  2730  2808  2886  2964  3042  3120  78 

79  2449  2528  2607  2686  2765  2844  2923  3002  3081  3160  79 

80  2480  2560  2640  2720  2800  2880  2960  3040  3120  3200  80 

81  2511  2592  2673  2754  2835  2916  2997  3078  3159  3240  81 

82  2542  2624  2706  2788  2870  2952  3034  3116  3198  3280  82 

83  2573  2656  2739  2822  2905  2988  3071  3154  3237  3320  83 

84  2604  2688  2772  2856  2940  3024  3108  3192  3276  3360  84 

85  2635  2720  2805  2890  2975  3060  3145  3230  3315  3400  85 

86  2666  2752  2838  2924  3010  3096  3182  3268  3354  3440  86 

87  2697  2784  2871  2958  3045  3132  3219  3306  3393  3480  87 

88  2728  2816  2904  2992  30S0  3168  325(1  :;:;it  3432  3520  88 

89  2759  2848  2937  3026  3115  3204  3293  3382  :*.  171  3560  89 

90  -'790  2880  2970  3060  3150  3240  3330  3420  3510  3600  90 


91  2821  2912  3003  3094  3185  3276  3367  3458  3519  3640  91 

92  2852  2944  3036  3128  :S22<>  3312  3404  ::i'.n;  :;  vss  3080  92 

93  2883  2976  3069  3162  3255  3348  3441  3534  3627  3720  93 

94  2914  3008  3102  3196  3290  3384  3478  3572  3666  .".760  94 

95  29 15  3040  3135  3230  3325  3420  3515  3610  3705  8800  95 

96  2976  3072  3168  3264  3360  3456  3552  3648  3744  3840  96 

97  3007  3104  3201  3298  3395  3492  3589  3686  3783  3880  97 

98  3038  3136  3234  3332  3130  3528  3626  3721  3822  3920  98 

99  3069  316s  3267  3366  3465  3564  3663  3762  3861  3960  99 
100  3100  3200  3300  3400  3500  3600  3700  3800  3900  4000  100 

31  32  33  34  35  36  37  38  39  40 


12 


178  MENTAL  AND  SOCIAL  MEASUREMENTS. 

41   42   43   *44   45   46   47   48   49   50 


1 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

1 

2 

82 

84 

86 

88- 

90 

92 

94 

96 

98 

100 

2 

3 

L23 

126 

129 

132 

135 

138 

141 

144 

147 

150 

3 

4 

164 

168 

172 

176 

180 

184 

188 

192 

196 

200 

4 

5 

205 

210 

215 

220 

225 

230 

235 

240 

245 

250 

5 

6 

246 

252 

258 

264 

270 

276 

282 

288 

294 

300 

6 

7 

287 

294 

301 

308 

315 

322 

329 

336 

343 

350 

7 

8 

328 

336 

344 

352 

360 

368 

376 

384 

392 

400 

8 

9 

369 

378 

387 

396 

405 

414 

423 

432 

441 

450 

9 

10 

410 

420 

430 

440 

450 

460 

470 

480 

490 

500 

10 

11 

451 

462 

473 

484 

495 

506 

517 

528 

539 

550 

11 

12 

492 

504 

516 

528 

540 

552 

564 

576 

588 

600 

12 

13 

533 

546 

559 

572 

585 

598 

611 

624 

637 

650 

13 

14 

574 

588 

602 

616 

630 

644 

658 

672 

686 

700 

14 

15 

615 

630 

645 

660 

675 

690 

705 

720 

735 

750 

15 

16 

656 

672 

688 

704 

720 

736 

752 

768 

784 

800 

16 

17 

697 

714 

731 

748 

765 

782 

799 

816 

833 

850 

17 

18 

738 

756 

774 

-792 

810 

828 

846 

864 

882 

900 

18 

19 

779 

798 

817 

836 

855 

874 

893 

912 

931 

950 

19 

20 

820 

840 

860 

880 

900 

920 

940 

960 

980 

1000 

20 

21 

861 

882 

903 

924 

945 

966 

987 

1008 

1029 

1050 

21 

22 

902 

924 

946 

968 

990 

1012 

1034 

1056 

1078 

1100 

22 

23 

943 

966 

989 

1012 

1035 

1058 

1081 

1104 

1127 

1150 

23 

24 

984 

1008 

1032 

1056 

1080 

1104 

1128 

1152 

1176 

1200 

24 

25 

1025 

1050 

1075 

1100 

1125 

1150 

1175 

1200 

1225 

1250 

25 

26 

1066 

1092 

1118 

1144 

1170 

1196 

1222 

1248 

1274 

1300 

26 

27 

1107 

1134 

1161 

1188 

1215 

1242 

1269 

1296 

1323 

1350 

27 

28 

1148 

1176 

1204 

1232 

1260 

1288 

1316 

1344 

1372 

1400 

28 

29 

1189 

1218 

1247 

1276 

1305 

1334 

1363 

1392 

1421 

1450 

29 

30 

1230 

1260 

1290 

1320 

1350 

1380 

1410 

1440 

1470 

1500 

30 

31 

1271 

1302 

1333 

1364 

1395 

1426 

1457 

1488 

1519 

1550 

31 

32 

1312 

1344 

1376 

1408 

1440 

1472 

1504 

1536 

1568 

1600 

32 

33 

1353 

1386 

1419 

1452 

1485 

1518 

1551 

1584 

1617 

1650 

33 

34 

1394 

1428 

1462 

1496 

1530 

1564 

1598 

1632 

1666 

1700 

34 

35 

1435 

1470 

1505 

1540 

1575 

1610 

1645 

1680 

1715 

1750 

35 

36 

1476 

1512 

1548 

1584 

1620 

1656 

1692 

1728 

1764 

1800 

36 

37 

1517 

1554 

1591 

1628 

1665 

1702 

1739 

1776 

1813 

1850 

37 

38 

1558 

1596 

1634 

1672 

1710 

1748 

1786 

1824 

1862 

1900 

38 

39 

1599 

1638 

1677 

1716 

1755 

1794 

1833 

1872 

1911 

1950 

39 

40 

1640 

1680 

1720 

1760 

1800 

1840 

1880 

1920 

1960 

2000 

40 

41 

1681 

1722 

1763 

1804 

1845 

1886 

1927 

1968 

2009 

2050 

41 

42 

1722 

1764 

1806 

1848 

1890 

1932 

1974 

2016 

2058 

2100 

42 

43 

1763 

1806 

1849 

1892 

1935 

1978 

2021 

2064 

2107 

2150 

43 

44 

1804 

1848 

1892 

1936 

1980 

2024 

2068 

2112 

2156 

2200 

44 

45 

1845 

1890 

1935 

1980 

2025 

2070 

2115 

2160 

2205 

2250 

45 

46 

1886 

1932 

1978 

2024 

2070 

2116 

2162 

2208 

2254 

2300 

46 

47 

1927 

1974 

2021 

2068 

2115 

2162 

2209 

2256 

2303 

2350 

47 

48 

1968 

2016 

2064 

2112 

2160 

2208 

2256 

2304 

2352 

2400 

48 

49 

2009 

2058 

2107 

2156 

2205 

2254 

2303 

2352 

2401 

2450 

49 

50 

2050 

2100 

2150 

2200 

2250 

2300 

2350 

2400 

2450 

2500 

50 

41   42   43   44   45   46   47   48   49   50 


A  MULTIPLICATION  TABLE.  179 

41  42  43  44  45  46  47  48  49   50 

51  2091  2142  2193  2244  229-5  2346  2397  2448  2499  2550  51 

52  2132  2184  2236  2288  2340  2392  2444  2496  2548  2600  52 

53  2173  2226  2279  2332  2385  2438  2491  2544  2597  2650  53 

54  2214  2268  2322  2376  2430  2484  2538  2592  2646  2700  54 

55  2255  2310  2365  2420  2475  2530  2585  2640  2695  2750  55 

56  2296  2352  2408  2464  2520  2576  2632  2688  2744  2800  56 

57  2337  2394  2451  2508  2565  2622  2679  2736  2793  2850  57 

58  2378  2436  2494  2552  2610  2668  2726  2784  2842  2900  58 

59  2419  2478  2537  2596  2655  2714  2773  2832  2891  2950  59 

60  2460  2520  2580  2640  2700  2760  2820  28S0  2940  3000  60 

61  2501  2562  2623  2684  2745  2806  2867  2928  2989  3050  61 

62  2542  2604  2666  2728  2790  2852  2914  2976  3038  3100  62 

63  2583  2646  2709  2772  2835  2898  2961  3024  3087  3150  63 

64  2624  2688  2752  2816  2880  2944  3008  3072  3136  3200  64 

65  2665  2730  2795  2860  2925  2990  3055  3120  3185  3250  65 

66  2706  2772  2838  2904  2970  3036  3102  3168  3234  3300  66 

67  2747  2814  2881  2948  3015  3082  3149  3216  3283  3350  67 

68  2788  2856  2924  2992  3060  3128  3196  3264  3332  3400  68 

69  2829  2898  2967  3036  3105  3174  3243  3312  3381  3450  69 

70  2870  2940  3010  3080  3150  3220  3290  3360  3430  3500  07 

71  2911  2982  3053  3124  3195  3266  3337  3408  3479  3550  71 

72  2952  3024  3096  3168  3240  3312  3384  3456  3528  3600  72 

73  2993  3066  3139  3212  3285  3358  3431  3504  3577  3650  73 

74  3034  3108  3182  3256  3330  3404  3478  3552  3626  3700  74 

75  3075  3150  3225  3300  3375  3450  3525  3600  3675  3750  75 

76  3116  3192  3268  3344  3420  3496  3572  3648  3724  3800  76 

77  3157  3234  3311  3388  3465  3542  3619  3696  3773  3850  77 

78  3198  3276  3354  3432  3510  3588  3666  3744  3822  3900  78 

79  3239  3318  3397  3476  3555  3634  3713  3792  3871  3950  79 

80  3280  3360  3440  3520  3600  3680  3760  3840  3920  4000  80 

81  3321  3402  3483  3564  3645  3726  3807  3888  3969  4050  81 

82  3362  3444  3526  3608  3690  3772  3854  3936  4018  41(10  82 

83  3403  3486  3569  3652  3735  3818  3901  3984  4067  4150  83 

84  3444  3528  3612  3696  3780  3864  3948  4032  4116  4200  84 

85  3485  3570  3655  3740  3825  3910  3995  40S0  4165  4250  85 

86  3526  3612  3698  3784  3870  3956  4042  4128  4214  4300  86 

87  3567  3654  3741  3828  3915  4002  4089  4176  4263  4350  87 

88  3608  3696  3784  -3872  3960  4048  4136  4224  4312  4400  88 

89  3649  3738  3827  3916  4005  4094  4183  4272  4361  4450  89 

90  3690  3780  3870  3960  4050  4140  4230  4320  4410  4500  90 

91  3731  3822  3913  4004  4095  4186  4277  4368  4459  4550  91 

92  3772  3864  3956  4048  4140  4232  4324  4416  4508  4600  92 

93  3813  3906  3999  4092  4185  4278  4371  4464  4557  4650  93 

94  3854  3948  4042  4136  4230  4324  4418  4512  4606  4700  94 

95  3895  3990  4085  4180  4275  4370  4465  1560  4655  4750  95 

96  3936  403-2  4128  4224  4320  4416  4512  4608  4704  4800  96 

97  3977  4074  4171  4268  4365  4462  4559  4656  -IT:).".  4850  97 

98  4018  4116  4'214  4312  4410  4508  1606  4704  4802  4900  98 

99  4059  4158  4257  4356  4455  4554  4653  4752  4851  4950  99 
100  4100  4200  4300  4400  4500  4600  4700  4800  4900  5000  100 

41  42   43  44  45   46   47  48  49   50 


180  MENTAL  AND  SOCIAL  MEASUREMENTS. 

51    52   53   54   55   56   57   58   59   60 


1 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

1 

2 

102 

104 

106 

108 

110 

112 

114 

116 

118 

120 

2 

3 

153 

156 

159 

162 

165 

168 

171 

174 

177 

180 

3 

4 

•Jul 

208 

212 

216 

220 

224 

228 

232 

236 

240 

4 

5 

255 

260 

265 

270 

275 

280 

285 

290 

295 

300 

5 

6 

306 

312 

318 

324 

330 

336 

342 

348 

354 

360 

6 

7 

357 

364 

371 

378 

385 

392 

399 

406 

413 

420 

7 

8 

408 

416 

424 

432 

440 

448 

456 

464 

472 

480 

8 

9 

459 

468 

477 

486 

495 

504 

513 

522 

531 

540 

9 

10 

510 

520 

530 

540 

550 

560 

570 

580 

590 

600 

10 

11 

561 

572 

583 

594 

605 

616 

627 

638 

649 

660 

11 

12 

612 

624 

636 

648 

660 

672 

684 

696 

708 

720 

12 

13 

663 

676 

689 

702 

715 

728 

741 

754 

767 

780 

13 

14 

714 

728 

742 

756 

770 

784 

798 

812 

826 

840 

14 

15 

765 

780 

795 

810 

825 

840 

855 

870 

885 

900 

15 

16 

816 

832 

848 

864 

880 

896 

912 

928 

944 

960 

16 

17 

867 

884 

901 

918 

935 

952 

969 

986 

1003 

1020 

17 

18 

918 

936 

954 

972 

990 

1008 

1026 

1044 

1062 

1080 

18 

19 

969 

988 

1007 

1026 

1045 

1064 

1083 

1102 

1121 

1140 

19 

20 

1020 

1040 

1060 

1080 

1100 

1120 

1140 

1160 

1180 

1200 

20 

21 

1071 

1092 

1113 

1134 

1155 

1176 

1197 

1218 

1239 

1260 

21 

22 

1122 

1144 

1166 

1188 

1210 

1232 

1254 

1276 

1298 

1320 

22 

23 

1173 

1196 

1219 

1242 

1265 

1288 

1311 

1334 

1357 

1380 

23 

24 

1224 

1248 

1272 

1296 

1320 

1344 

1368 

1392 

1416 

1440 

24 

25 

1275 

1300 

1325 

1350 

1375 

1400 

1425 

1450 

1475 

1500 

25 

26 

1326 

1352 

1378 

1404 

1430 

1456 

1482 

1508 

1534 

1560 

26 

27 

1377 

1404 

1431 

1458 

1485 

1512 

1539 

1566 

1593 

1620 

27 

28 

1428 

1456 

1484 

1512 

1540 

1568 

1596 

1624 

1652 

1680 

28 

29 

1479 

1508 

1537 

1566 

1595 

1624 

1653 

1682 

1711 

1740 

29 

30 

1530 

1560 

1590 

1620 

1650 

1680. 

1710 

1740 

1770 

1800 

30 

31 

1581 

1612 

1643 

1674 

1705 

1736 

1767 

1798 

1829 

1860 

31 

32 

1632 

1664 

1696 

1728 

1760 

1792 

1824 

1856 

1888 

1920 

32 

33 

1683 

1716 

1749 

1782 

1815 

1848 

1881 

1914 

1947 

1980 

33 

34 

1734 

1768 

1802 

1836 

1870 

1904 

1938 

1972 

2006 

2040 

34 

35 

1785 

1820 

1855 

1890 

1925 

1960 

1995 

2030 

2065 

2100 

35 

36 

1836 

1872 

1908 

1944 

1980 

2016 

2052 

2088 

2124 

2160 

36 

37 

1887 

1924 

1961 

1998 

2035 

2072 

2109 

2146 

2183 

2220 

37 

38 

1938 

1976 

2014 

2052 

2090 

2128 

2166 

2204 

2242 

2280 

38 

39 

1989 

2028 

2067 

2106 

2145 

2184 

2223 

2262 

2301 

2340 

39 

40 

2040 

2080 

2120 

2160 

2200 

2240 

2280 

2320 

2360 

2400 

40 

41 

2091 

2132 

2173 

2214 

2255 

2296 

2337 

2378 

2419 

2460 

41 

42 

2142 

2184 

2226 

2268 

2310 

2352 

2394 

2436 

2478 

2520 

42 

43 

2193 

2236 

2279 

2322 

2365 

2408 

2451 

2494 

2537 

2580 

43 

44 

2244 

2288 

2332 

2376 

2420 

2464 

2508 

2552 

2596 

2640 

44 

45 

2295 

2340 

2385 

2430 

2475 

2520 

2565 

2610 

2655 

2700 

45 

46 

2346 

2392 

2438 

2484 

2530 

2576 

2622 

2668 

2714 

2760 

46 

47 

2397 

2444 

2491 

2538 

2585 

2632 

2679 

2726 

2773 

2820 

47 

48 

2448 

2496 

2544 

2592 

2640 

2688 

2736 

2784 

2832 

2880 

48 

49 

2499 

2548 

2597 

2646 

2695 

2744 

2793 

2842 

2891 

2940 

49 

50 

2550 

2600 

2650 

2700 

2750 

2800 

2850 

2900 

2950 

3000 

50 

51    52   53   54   55   56   57    58   59   60 


A  MULTIPLICATION  TABLE.  181 

51  52   53  54  55  56  57  58  59   60 

51  2601  2652  2703  2754  2805  2856  2907  2958  3009  3060  51 

52  2652  2704  2756  2808  2860  2912  2964  3016  3068  3120  52 

53  2703  2756  2809  2862  2915  2968  3021  3074  3127  3180  53 

54  2754  2808  2862  2916  2970  3024  3078  3132  3186  3240  54 

55  2805  2860  2915  2970  3025  3080  3135  3190  3245  3300  55 

56  2856  2912  2968  3024  3080  3136  3192  3248  3304  3360  56 

57  2907  2964  3021  3078  3135  3192  3249  3306  3363  3420  57 

58  2958  3016  3074  3132  3190  3248  3306  3364  3422  3480  58 

59  3009  3068  3127  3186  3245  3304  3363  3422  3481  3540  59 

60  3060  3120  3180  3240  3300  3360  3420  3480  3540  3600  60 

61  3111  3172  3233  3294  3355  3416  3477  3538  3599  3660  61 

62  3162  3224  3286  3348  3410  3472  3534  3596  3658  372o  62 

63  3213  3276  3339  3402  3465  3528  3591  3654  3717  3780  63 

64  3264  3328  3392  3456  3520  3584  3648  3712  3776  3840  64 

65  3315  3380  3445  3510  3575  3640  3705  3770  3835  3900  65 

66  3366  3432  3498  3564  3630  3696  3762  3828  3894  3960  66 

67  3417  3484  3551  3618  3685  3752  3819  3886  3953  4020  67 

68  3468  3536  3604  3672  3740  3808  3876  3944  4012  4080  68 

69  3519  3588  3657  3726  3795  3864  3933  4002  4071  4140  69 

70  3570  3640  3710  3780  3850  3920  3990  4060  4130  4200  70 

71  3621  3692  3763  3834  3905  3976  4047  4118  4189  4260  71 

72  3672  3744  3816  3888  3960  4032  4104  4176  4248  4320  72 

73  3723  3796  3869  3942  4015  4088  4161  4234  4307  4380  73 

74  3774  3848  3922  3996  4070  4144  4218  4292  4366  4440  74 

75  3825  3900  3975  4050  4125  4200  4275  4350  4425  4500  75 

76  3876  3952  4028  4104  4180  4256  4332  4408  4484  4560  76 

77  3927  4004  4081  4158  4235  4312-  4389  4466  4543  4620  77 

78  3978  4056  4134  4212  4290  4368  4446  4524  4602  4680  78 

79  4029  4108  4187  4266  4345  4424  4503  4582  4661  4740  79 

80  4080  4160  4240  4320  4400  4480  4560  4640  4720  4800  80 


81  4131  4212  4293  4374  4455  4536  4617  4698  4779  4860  81 

82  4182  4264  4346  4428  4510  4592  4674  4756  4838  4920  82 

83  4233  4316  4399  4482  4565  4648  4731  4814  4897  4980  83 

84  4284  4368  4452  4536  4620  4704  4788  4872  495K  5040  84 

85  4335  4420  4505  4590  4675  4760  -is  IT,  4930  5015  5100  85 

86  4386  4472  455S  464-1  4730  4S16  4902  49XS  5074  5160  86 

87  4437  4524  4611  4698  4785  4872  4959  5046  5133,  5220  87 

88  4488  4576  4664  4752  4S40  4928  5016  5104  5192  5280  88 

89  4539  1628  4717  4806  4895  4984  5073  5162  5251  5340  89 

90  4590  4680  4770  4860  4950  5040  5130  5220  5310  5400  90 

91  4641  4732  4823  4914  5005  5096  5187  5278  5369  5160  91 

92  4692  4784  4876  4968  5060  5152  5244  5336  5428  5520  92 

93  4743  4836  4929  5022  5115  5208  5301  5394  5487  5580  93 

94  4794  4888  4982  5076  5170  5264  5358  5  152  55  16  56 10  94 

95  4845  4940  5035  5130  5225  5320  5415  5510  5605  5700  95 

96  4896  1992  5088  5184  52S0  5376  5172  5568  5664  5760  96 

97  19  17  5044  5141  5238  5335  5132  5529  5626  5723,  5820  97 

98  499S  5096  519-1  5292  53,1)0  5488  5586  5684  5782  5880  98 

99  5049  5148  5217  5346  5445  5544  56  13,  57  12  5841  59  lo  99 
100  5100  5200  5300  5400  5500  5600  5700  5800  5900  6000  100 

51  52  53  54  55  56  57  58  59  60 


182  MENTAL  AND  SOCIAL  MEASUREMENTS. 

61   62   63   64   65   66   67   68   69   70 


1 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

1 

2 

122 

124 

126 

128 

130 

132 

134 

136 

138 

140 

2 

3 

183 

186 

189 

192 

195 

198 

201 

204 

207 

210 

3 

4 

•J  1 1 

218 

252 

256 

260 

264 

268 

272 

276 

280 

4 

5 

305 

310 

315 

320 

325 

330 

335 

340 

345 

350 

5 

6 

366 

372 

378 

384 

390 

396 

402 

408 

414 

420 

6 

7 

427 

434 

441 

448 

455 

462 

469 

476 

483 

490 

7 

8 

488 

496 

504 

512 

520 

528 

536 

544 

552 

560 

8 

9 

549 

558 

567 

576 

585 

594 

603 

612 

621 

630 

9 

10 

610 

620 

630 

640 

650 

660 

670 

680 

690 

700 

10 

11 

671 

682 

693 

704 

715 

726 

737 

748 

759 

770 

11 

12 

732 

744 

756 

768 

780 

792 

804 

816 

828 

840 

12 

13 

793 

806 

819 

832 

845 

858 

871 

884 

897 

910 

13 

14 

854 

868 

882 

896 

910 

924 

938 

952 

966 

980 

14 

15 

915 

930 

945 

960 

975 

990 

1005 

1020 

1035 

1050 

15 

16 

976 

992 

1008 

1024 

1040 

1056 

1072 

1088 

1104 

1120 

16 

17 

1037 

1054 

1071 

1088 

1105 

1122 

1139 

1156 

1173 

1190 

17 

18 

1098 

1116 

1134 

1152 

1170 

1188 

1206 

1224 

1242 

1260 

18 

19 

1159 

1178 

1197 

1216 

1235 

1254 

1273 

1292 

1311 

1330 

19 

20 

1220 

1240 

1260 

1280 

1300 

1320 

1340 

1360 

1380 

1400 

20 

21 

1281 

1302 

1323 

1344 

1365 

1386 

1407 

1428 

1449 

1470 

21 

22 

1342 

1364 

1386 

1408 

1430 

1452 

1474 

1496 

1518 

1540 

22 

23 

1403 

1426 

1449 

1472 

1495 

1518 

1541 

1564 

1587 

1610 

23 

24 

1464 

1488 

1512 

1536 

1560 

1584 

1608 

1632 

1656 

1680 

24 

25 

1525 

1550 

1575 

1600 

1625 

1650 

1675 

1700 

1725 

1750 

25 

26 

1586 

1612 

1638 

1664 

1690 

1716 

1742 

1768 

1794 

1820 

26 

27 

1647 

1674 

1701 

1728 

1755 

1782 

1809 

1836 

1863 

1890 

27 

28 

1708 

1736 

1764 

1792 

1820 

1848 

1876 

1904 

1932 

1960 

28 

29 

1769 

1798 

1827 

1856 

1885 

1914 

1943 

1972 

2001 

2030 

29 

30 

1830 

1860 

1890 

1920 

1950 

1980 

2010 

2040 

2070 

2100 

30 

31 

1891 

1922 

1953 

1984 

2015 

2046 

2077 

2108 

2139 

2170 

31 

32 

1952 

1984 

2016 

2048 

2080 

2112 

2144 

2176 

2208 

2240 

32 

33 

2013 

2046 

2079 

2112 

2145 

2178 

2211 

2244 

2277 

2310 

33 

34 

2074 

2108 

2142 

2176 

2210 

2244 

2278 

2312 

2346 

2380 

34 

35 

2135 

2170 

2205 

2240 

2275 

2310 

2345 

2380 

2415 

2450 

35 

36 

2196 

2232 

2268 

2304 

2340 

2376 

2412 

2448 

2484 

2520 

36 

37 

2257 

2294 

2331 

2368 

2405 

2442 

2479 

2516 

2553 

2590 

37 

38 

2318 

2356 

2394 

2432 

2470 

2508 

2546 

2584 

2622 

2660 

38 

39 

2379 

2418 

2457 

2496 

2535 

2574 

2613 

2652 

2691 

2730 

39 

40 

2440 

2480 

2520 

2560 

2600 

2640 

2680 

2720 

2760 

2800 

40 

41 

2501 

2542 

2583 

2624 

2665 

2706 

2747 

2788 

2829 

2870 

41 

42 

2562 

2604 

2646 

2688 

2730 

2772 

2814 

2856 

2898 

2940 

42 

43 

2623 

2666 

2709 

2752 

2795 

2838 

2881 

2924 

2967 

3010 

43 

44 

2684 

2728 

2772 

2816 

2860 

2904 

2948 

2992 

3036 

3080 

44 

45 

2745 

2790 

2835 

2880 

2925 

2970 

3015 

3060 

3105 

3150 

45 

46 

2806 

2852 

2898 

2944 

2990 

3036 

3082 

3128 

3174 

3220 

46 

47 

2867 

2914 

2961 

3008 

3055 

3102 

3149 

3196 

3243 

3290 

47 

48 

2928 

2976 

3024 

3072 

3120 

3168 

3216 

3264 

3312 

3360 

48 

49 

2989 

3038 

3087 

3136 

3185 

3234 

3283 

3332 

3381 

3430 

49 

50 

3050 

3100 

3150 

3200 

3250 

3300 

3350 

3400 

3450 

3500 

50 

61   62   63   64   65   66   67   68   69   70 


A  MULTIPLICATION  TABLE.  183 

61  62  63  64  65   66  67   68   69  70 

51  3111  3162  3213  3264  3315  3366  3417  3468  3519  3570  51 

52  3172  3224  3276  3328  3380  3432  3484  3536  3588  3640  52 

53  3233  3286  3339  3392  3445  3498  3551  3604  3657  3710  53 

54  3294  3348  3402  3456  3510  3564  3618  3672  3726  3780  54 

55  3355  3410  3465  3520  3575  3630  3685  3740  3795  3850  55 

56  3416  3472  3528  3584  3640  3696  3752  3808  3864  3920  56 

57  3477  3534  3591  3648  3705  3762  3819  3876  3933  3990  57 

58  3538  3596  3654  3712  3770  3828  3886  3944  4002  4060  58 

59  3599  3658  3717  3776  3835  3894  3953  4012  4071  4130  59 

60  3660  3720  3780  3840  3900  3960  4020  4080  4140  4200  60 

61  3721  3782  3843  3904  3965  4026  4087  4148  4209  4270  61 

62  3782  3844  3906  3968  4030  4092  4154  4216  4278  4340  62 

63  3843  3906  3969  4032  4095  4158  4221  4284  4347  4410  63 

64  3904  3968  4032  4096  4160  4224  4288  4352  4416  4480  64 

65  3965  4030  4095  4160  4225  4290  4355  4420  4485  4550  65 

66  4026  4092  4158  4224  4290  4356  4422  4488  4554  4620  66 

67  4087  4154  4221  4288  4355  4422  4489  4556  4623  4690  67 

68  4148  4216  4284  4352  4420  4488  4556  4624  4692  4760  68 

69  4209  4278  4347  4416  4485  4554  4623  4692  4761  4830  69 

70  4270  4340  4410  4480  4550  4620  4690  4760  4830  4900  70 

71  4331  4402  4473  4544  4615  4686  4757  4828  4899  4970  71 

72  4392  4464  4536  4608  4680  4752  4824  4896  4968  5040  72 

73  4453  4526  4599  4672  4745  4818  4891  4964  5037  5110  73 

74  4514  4588  4662  4736  4810  4884  4958  5032  5106  5180  74 

75  4575  4650  4725  4800  4875  4950  5025  5100  5175  5250  75 

76  4636  4712  4788  4864  4940  5016  5092  5168  5244  5320  76 

77  4697  4774  4851  4928  5005  5082  5159  5236  5313  5390  77 

78  4758  4836  4914  4992  5070  5148  5226  5304  5382  5460  78 

79  4819  4898  4977  5056  5135  5214  5293  5372  5451  5530  79 

80  4880  4960  5040  5120  5200  5280  5360  5440  5520  5600  80 

81  4941  5022  5103  5184  5265  5346  5427  5508  5589  5670  81 

82  5002  5084  5166  5248  5330  5412  5494  5576  5658  5740  82 

83  5063  5146  5229  5312  5395  5478  5561  5644  5727  5810  83 

84  5124  5208  5292  5376  5460  5544  5628  5712  5796  5880  84 

85  5185  5270  5355  5440  5525  5610  5695  5780  5865  5950  85 

86  5246  5332  5418  5504  5590  5676  5762  5848  5934  6020  86 

87  5307  5394  5481  5568  5655  5742  5829  5916  6003  6090  87 

88  5368  5456  5544  5632  5720  5808  5896  5934  6072  6160  88 

89  5429  5518  5607  5696  5785  5874  5963  6052  6141  6230  89 

90  5490  5580  5670  5760  5850  5940  6030  6120  6210  6300  90 

91  5551  5642  5733  5824  5915  6006  6097  6188  6279  6370  91 

92  5612  5704  5796  5888  5980  6072  6164  6256  0348  6440  92 

93  5673  5766  5859  5952  6045  6138  6231  6324  6417  6510  93 

94  5734  5828  5922  6016  6110  6204  6298  6392  6486  6580  94 

95  5795  5890  5985  6080  6175  6270  6365  6460  6555  6650  95 

96  5856  5952  6048  6144  6240  6336  6432  0528  00\M  0720  96 

97  5917  6014  6111  6208  6305  6402  6499  6596  6693  6790  97 

98  5978  6076  6174  6272  6370  6468  6566  6664  6762  6800  98 

99  6039  6138  6237  0336  6435  6534  6633  6732  6831  6930  99 
100  0100  6200  6300  6400  6500  6600  0700  6800  6900  7000  100 

61  62  63  64  65   66  67   68   69  70 


184  MENTAL    AND  SOCTAL  MEASUREMENTS. 

71    72   73    74   75   76   77    78   79   80 


1 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

1 

2 

142 

144 

146 

148 

L50 

152 

154 

156 

158 

160 

2 

3 

213 

216 

219 

222 

225 

228 

231 

234 

237 

240 

3 

4 

28 1 

288 

292 

296 

300 

304 

308 

312 

316 

320 

4 

5 

355 

360 

365 

370 

375 

380 

385 

390 

395 

400 

5 

6 

426 

432 

438 

444 

450 

456 

462 

468 

474 

480 

6 

7 

497 

504 

511 

518 

525 

532 

539 

546 

553 

560 

7 

8 

568 

576 

584 

592 

600 

608 

616 

624 

632 

640 

8 

9 

639 

648 

657 

666 

675 

684 

693 

702 

711 

720 

9 

10 

710 

720 

730 

740 

750 

760 

770 

780 

790 

800 

10 

11 

781 

792 

803 

814 

825 

836 

847 

858 

869 

880 

11 

12 

852 

864 

876 

888 

900 

912 

924 

936 

948 

960 

12 

13 

923 

936 

949 

962 

975 

988 

1001 

1014 

1027 

1040 

13 

14 

994 

1008 

1022 

1036 

1050 

1064 

1078 

1092 

1106 

1120 

14 

15 

1065 

1080 

1095 

1110 

1125 

1140 

1155 

1170 

1185 

1200 

15 

16 

1136 

1152 

1168 

1184 

1200 

1216 

1232 

1248 

1264 

1280 

16 

17 

1207 

1224 

1241 

1258 

1275 

1292 

1309 

1326 

1343 

1360 

17 

18  • 

1278 

1296 

1314 

1332 

1350 

1368 

1386 

1404 

1422 

1440 

18 

19 

1349 

1368 

1387 

1406 

1425 

1444 

1463 

1482 

1501 

1520 

19 

20 

1420 

1440 

1160 

1480 

1500 

1520 

1540 

1560 

1580 

1600 

20 

21 

1491 

1512 

1533 

1554 

1575 

1596 

1617 

1638 

1659 

1680 

21 

22 

1562 

1584 

1606 

1628 

1650 

1672 

1694 

1716 

1738 

1760 

22 

23 

1633 

1656 

1679 

1702 

1-725 

1748 

1771 

1794 

1817 

1840 

23 

24 

1704 

1728 

1752 

1776 

1800 

1824 

1848 

1872 

1896 

1920 

24 

25 

1775 

1800 

1825 

1850 

1875 

1900 

1925 

J  950 

1975 

2000 

25 

26 

1846 

1872 

1898 

1924 

1950 

1976 

2002 

2028 

2054 

2080 

26 

27 

1917 

1944 

1971 

1998 

2025 

2052 

2079 

2106 

2133 

2160 

27 

28 

1988 

2016 

2044 

2072 

2100 

2128 

2156 

2184 

2212 

2240 

28 

29 

2059 

2088 

2117 

2146 

2175 

2204 

2233 

2262 

2291 

23:0 

29 

30 

2130 

2160 

2190 

2220 

2250 

2280 

2310 

2340 

2370 

2400 

30 

31 

2201 

2232 

2263 

2294 

2325 

2356 

2387 

2418 

2449 

2480 

31 

32 

2272 

2304 

2336 

2368 

2400 

2432 

2464 

2496 

2528 

2560 

32 

33 

2343 

2376 

2409 

2442 

2475 

2508 

2541 

2574 

2607 

2640 

33 

34 

2414 

2448 

2482 

2516 

2550 

2584 

2618 

2652 

2686 

2720 

34 

35 

2485 

2520 

2555 

2590 

2625 

2660 

2695 

2730 

2765 

2800 

35 

36 

2556 

2592 

2628 

2664 

2700 

2736 

2772 

2808 

28i4 

2880 

36 

37 

2627 

2664 

2701 

2738 

2775 

2812 

2849 

2886 

2923 

2960 

37 

38 

2698 

2736 

2774 

2812 

2850 

2888 

2926 

2964 

3002 

3040 

38 

39 

2769 

2808 

2847 

2886 

2925 

2964 

3003 

3<»42 

3081 

3120 

39 

40 

2840 

2880 

2920 

2960 

3000 

3040 

3080 

3120 

3160 

3200 

40 

41 

2911 

2952 

2993 

3034 

3075 

3116 

3157 

3198 

3239 

3280 

41 

42 

2982 

3024 

3066 

3108 

3150 

3192 

3234 

3276 

3318 

3360 

42 

43 

3053 

3096 

3139 

3182 

3225 

3268 

3311 

3354 

3397 

3440 

43 

44 

3124 

3168 

3212 

3256 

3300 

3344 

3388 

3432 

3476 

3520 

44 

45 

3195 

3240 

3285 

3330 

3375 

3420 

3465 

3510 

3555 

3600 

45 

46 

3266 

3312 

3358 

3404 

3450 

3496 

3542 

3588 

3634 

3680 

46 

47 

3337 

3384 

3431 

3478 

3525 

3572 

3619 

3666 

3713 

3760 

47 

48 

3408 

3456 

3504 

3552 

3600 

3648 

3696 

3744 

3792 

3840 

48 

49 

3479 

3528 

3577 

3626 

3675 

3724 

3773 

3822 

3871 

3920 

49 

50 

3550 

3600 

3650 

3700 

3750 

3800 

3850 

3900 

3950 

4000 

50 

71   72   73   74   75   76   77   78   79   80 


A  MULTIPLICATION  TABLE.  185 

71  72    73  74  75   76    77  78  79  80 

51  3621  3672  3723  3774  3825  3876  3927  3978  4029  4080  51 

52  3692  3744  3796  3848  3900  3952  4004  4056  4108  4160  52 

53  3763  3816  3869  3922  3975  4028  4081  4134  4187  4240  53 

54  3834  3888  3942  3996  4050  4104  4158  4212  4266  4320  54 

55  3905  3960  4015  4070  4125  4180  4235  4290  4345  4400  55 

56  3976  4032  4088  4144  4200  4256  4312  4368  4424  4480  56 

57  4047  4104  4161  4218  4275  4332  4389  4446  4503  4560  57 

58  4118  4176  4234  4292  4350  4408  4466  4524  4582  4640  58 

59  4189  4248  4307  4366  4425  4484  4543  4602  4661  4720  59 

60  4260  4320  4380  4440  4500  4560  4620  4680  4740  4800  60 

61  4331  4392  4453  4514  4575  4636  4697  4758  4819  4880  61 

62  4402  4464  4526  4588  4650  4712  4774  4836  4898  4960  62 

63  4473  4536  4599  4662  4725  4788  4851  4914  4977  5040  63 

64  4544  4608  4672  4736  4800  4864  4928  4992  5056  5120  64 

65  4615  4680  4745  4810  4875  4940  5005  5070  5135  5200  65 

66  4686  4752  4818  4884  4950  5016  5082  5148  5214  5280  66 

67  4757  4824  4891  4958  5025  5092  5159  5226  5293  5360  67 

68  4828  '4896  4964  5032  5100  5168  5236  5304  5372  5440  68 

69  4899  4968  5037  5106  5175  5244  5313  5382  5451  5520  69 

70  4970  5040  5110  5180  5250  5320  5390  5460  5530  5600  70 

71  5041  5112  5183  5254  5325  5396  5467  5538  5609  5680  71 

72  5112  5184  5256  5328  5400  5472  5544  5616  5688  5760  72 

73  5183  5256  5329  5402  5475  5548  5621  5694  5767  5840  73 

74  5254  5328  5402  5476  5550  5624  5698  5772  5846  5920  74 

75  5325  5400  5475  5550  5625  5700  5775  5850  5925  6000  75 

76  5396  5472  5548  5624  5700  5776  5852  5928  6004  6080  76 

77  5467  5544  5621  5698  5775  5852  5929  6006  6083  6160  77 

78  5538  5616  5694  5772  5850  5928  6006  6084  6162  6240  78 

79  5609  5688  5767  5846  5925  6004  6083  6162  6241  6320  79 

80  5680  5760  5840  5920  6000  6080  6160  6240  6320  6400  80 

81  5751  5832  5913  5994  6075  6156  6237  6318  6399  6480  81 

82  5822  5904  5986  6068  6150  6232  6314  6396  6478  6560  82 

83  5893  5976  6059  6142  6225  6308  6391  6474  6557  6640  83 

84  5964  6048  6132  6216  6300  6384  6468  6552  6636  6720  84 

85  6035  6120  6205  6290  6375  6460  6545  6630  6715  6800  85 

86  6106  6192  6278  6364  6450  6536  6622  6708  6794  6880  86 

87  6177  6264  6351  6438  6525  6612  6699  6786  6873  6960  87 

88  6248  6336  6424  6512  6600  6688  6776  6864  6952  7040  88 

89  6319  6408  6497  6586  6675  6764  6853  6942  7031  7120  89 

90  6390  6480  6570  6660  6750  6840  6930  7020  7110  7200  90 

91  6461  6552  6643  6734  6825  6916  7007  7098  7189  7280  91 

92  6532  6624  6716  6808  6900  6992  7084  7176  7268  7360  92 

93  6603  6696  6789  6882  6975  7068  7161  7254  7347  7440  93 

94  6674  6768  6862  6956  7050  7144  7238  7332  7426  7520  94 

95  6745  6H40  6935  7030  7125  7220  7315  7410  7505  7600  95 

96  6816  6912  7008  7104  7200  7296  7392  7488  7581  7680  96 

97  6887  6984  7081  7178  7275  7372  7469  7.r»(i(i  7663  7760  97 

98  6958  7056  7154  7252  7350  7448  7546  7644  7742  7840  98 

99  7029  7128  7227  7326  7425  7524  7623  7722  7821  7920  99 
100  7100  7200  7300  7100  7500  7600  7700  7800  7900  8000  100 

71  72  73  74  75   76    77  78  79  80 


186  MENTAL  AND  SOCIAL  MEASUREMENTS. 

81   82   83   84   85   86   87   88   89   90 


1 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

1 

2 

162 

164 

166 

168 

170 

172 

174 

176 

178 

180 

2 

3 

243 

246 

249 

252 

255 

258 

261 

264 

267 

270 

3 

4 

324 

328 

332 

336 

340 

344 

348 

352 

356 

360 

4 

5 

405 

410 

415 

420 

425 

430 

435 

440 

445 

450 

5 

6 

486 

492 

498 

504 

510 

516 

522 

528 

534 

540 

6 

7 

567 

574 

581 

588 

595 

602 

609 

616 

623 

630 

7 

8 

648 

656 

664 

672 

680 

688 

696 

704 

712 

720 

8 

9 

729 

738 

747 

756 

765 

774 

783 

792 

801 

810 

9 

10 

810 

820 

830 

840 

850 

860 

870 

880 

890 

900 

10 

11 

891 

902 

913 

924 

935 

946 

957 

968 

979 

990 

11 

12 

972 

984 

996 

1008 

1020 

1032 

1044 

1056 

1068 

1080 

12 

13 

1053 

1066 

1079 

1092 

1105 

1118 

1131 

1144 

1157 

1170 

13 

14 

1134 

1148 

1162 

1176 

1190 

1204 

1218 

1232 

1246 

1260 

14 

15 

1215 

1230 

1245 

1260 

1275 

1290 

1305 

1320 

1335 

1350 

15 

16 

1296 

1312 

1328 

1344 

1360 

1376 

1392 

1408 

1424 

1440 

16 

17 

1377 

1394 

1411 

1428 

1445 

1462 

1479 

1496 

1513 

1530 

17 

18 

1458 

1476 

1494 

1512 

1530 

1548 

1566 

1584 

1602 

1620 

18 

19 

1539 

1558 

1577 

1596 

1615 

1634 

1653 

1672 

1691 

1710 

19 

20 

1620 

1640 

1660 

1680 

1700 

1720 

1740 

1760 

1780 

1800 

20 

21 

1701 

1722 

1743 

1764 

1785 

1806 

1827 

1848 

1869 

1890 

21 

22 

1782 

1804 

1826 

1848 

1870 

1892 

1914 

1936 

1958 

1980 

22 

23 

1863 

1886 

1909 

1932 

1955 

1978 

2001 

2024 

2047 

2070 

23 

24 

1944 

1968 

1992 

2016 

2040 

2064 

2088 

2112 

2136 

2160 

24 

25 

2025 

2050 

2075 

2100 

2125 

2150 

2175 

2200 

2225 

2250 

25 

26 

2106 

2132 

2158 

2184 

2210 

2236 

2262 

2288 

2314 

2340 

26 

27  ' 

2187 

2214 

2241 

2268 

2295 

2322 

2349 

2376 

2403 

2430 

27 

28 

2268 

2296 

2324 

2352 

2380 

2408 

2436 

2464 

2492 

2520 

28 

29 

2349 

2378 

2407 

2436 

2465 

2494 

2523 

2552 

2581 

2610 

29 

30 

2430 

2460 

2490 

2520 

2550 

2580 

2610 

2640 

2670 

2700 

30 

31 

2511 

2542 

2573 

2604 

2635 

2666 

2697 

2728 

2759 

2790 

31 

32 

2592 

2624 

2656 

2688 

2720 

2752 

2784 

2816 

2848 

2880 

32 

33 

2673 

2706 

2739 

2772 

2805 

2838 

2871 

2904 

2937 

2970 

33 

34 

2754 

2788 

2822 

2856 

2890 

2924 

2958 

2992 

3026 

3060 

34 

35 

2835 

2870 

2905 

2940 

2975 

3010 

3045 

3080 

3115 

3150 

35 

36 

2916 

2952 

2988 

3024 

3060 

3096 

3132 

3168 

3204 

3240 

36 

37 

2997 

3034 

3071 

3108 

3145 

3182 

3219 

3256 

3293 

3330 

37 

38 

3078 

3116 

3154 

3192 

3230 

3268 

3306 

3344 

3382 

3420 

38 

39 

3159 

3198 

3237 

3276 

3315 

3354 

3393 

3432 

3471 

3510 

39 

40 

3240 

3280 

3320 

3360 

3400 

3440 

3480 

3520 

3560 

3600 

40 

41 

3321 

3362 

3403 

3444 

3485 

3526 

3567 

3608 

3649 

3690 

41 

42 

3402 

3444 

3486 

3528 

3570 

3612 

3654 

3696 

3738 

3780 

42 

43 

3483 

3526 

3569 

3612 

3655 

3698 

3741 

3784 

3827 

3870 

43 

44 

3564 

3606 

3652 

3696 

3740 

3784 

3828 

3872 

3916 

3960 

44 

45 

3645 

3690 

3735 

3780 

3825 

3870 

3915 

3960 

4005 

4050 

45 

46 

3726 

3772 

3818 

3864 

3910 

3956 

4002 

4048 

4094 

4140 

46 

47 

3807 

3854 

3901 

3948 

3995 

4042 

4089 

4136 

4183 

4230 

47 

48 

3888 

3936 

3984 

4032 

4080 

4128 

4176 

4224 

4272 

4320 

48 

49 

3969 

4018 

4067 

4116 

4165 

4214 

4263 

4312 

4361 

4410 

49 

50 

4050 

4100 

4150 

4200 

4250 

4300 

4350 

4400 

4450 

4500 

50 

81   82   83   84   85   86   87   88   89   90 


A  MULTIPLICATION  TABLE.  187 

81  82  83  84    85   86  87  88  89  90 

51  4131  4182  4233  4284  4335  4386  4437  4488  4539  4590  51 

52  4212  4264  4316  4368  4420  4472  4524  4576  4628  4680  52 

53  4293  4346  4399  4452  4505  4558  4611  4664  4717  4770  53 

54  4374  4428  4482  4536  4590  4644  4698  4752  4806  4860  54 

55  4455  4510  4565  4620  4675  4730  4785  4840  4895  4950  55 

56  4536  4592  4648  4704  4760  4816  4872  4928  4984  5040  56 

57  4617  4674  4731  4788  4845  4902  4959  5016  5073  5130  57 

58  4698  4756  4814  4872  4930  4988  5046  5104  5162  5220  58 

59  4779  4838  4897  4956  5015  5074  5133  5192  5251  5310  59 

60  4860  4920  4980  5040  5100  5160  5220  5280  5340  5400  60 

61  4941  5002  5063  5124  5185  5246  5307  5368  5429  5490  61 

62  5022  5084  5146  5208  5270  5332  5394  5456  5518  5580  62 

63  5103  5166  5229  5292  5355  5418  5481  5544  5607  5670  63 

64  5184  5248  5312  5376  5440  5504  5568  5632  5696  5760  64 

65  5265  5330  5395  5460  5525  5590  5655  5720  5785  5850  65 

66  5346  5412  5478  5544  5610  5676  5742  5808  5874  5940  66 

67  5427  5494  5561  5628  5695  5762  5829  5896  5963  6030  67 

68  5508  5576  5644  5712  5780  5848  5916  5984  6052  6120  68 

69  5589  5658  5727  5796  5865  5934  6003  6072  6141  6210  89 

70  5670  5740  5810  5880  5950  6020  6090  6160  6230  6300  70 

71  5751  5822  5893  5964  6035  6106  6177  6248  6319  6390  71 

72  5832  5904  5976  6048  6120  6192  6264  6336  6408  6480  72 

73  5913  5986  6059  6132  6205  6278  6351  6424  6497  6570  73 

74  5994  6068  6142  6216  6290  6364  6438  6512  6586  6660  74 

75  6075  6150  6225  6300  6375  6450  6525  6600  6675  6750  75 

76  6156  6232  6308  6384  6460  6536  6612  6688  6764  6840  76 

77  6237  6314  6391  6468  6545  6622  6699  6776  6853  6930  77 

78  6318  6396  6474  6552  6630  6708  6786  6864  6942  7020  78 

79  6399  6478  6557  6636  6715  6794  6873  6952  7031  7110  79 

80  6480  6560  6640  6720  6800  6880  6960  7040  7120  7200  80 

81  6561  6642  6723  6804  6885  6966  7047  7128  7209  7290  81 

82  6642  6724  6806  6888  6970  7052  7134  7216  7298  7380  82 

83  6723  6806  6889  6972  7055  7138  7221  7304  7387  7470  83 

84  6804  6888  6972  7056  7140  7224  7308  7392  7476  7560  84 

85  6H85  6970  7055  7140  7225  7310  7395  7480  7565  7650  85 

86  6966  7052  7138  7224  7310  7396  7482  7568  7654  7740  86 

87  7047  7134  7221  7308  7395  7482  7569  7656  7743  7830  87 

88  7128  7216  7304  7392  7480  7568  7656  7744  7832  7920  88 

89  7209  7298  7387  7476  7565  7654  7743  7832  7921  8010  89 

90  7290  7380  7470  7560  7650  7740  7830  7920  8010  8100  90 

91  7371  7462  7553  7644  7735  7826  7917  8008  8099  8190  91 

92  7452  7544  7636  7728  7820  7912  8004  8096  8188  8280  92 

93  7533  7626  7719  7812  7905  7998  8091  8184  8277  8370  93 

94  7614  7708  7802  7896  7990  8084  8178  8272  8366  8460  94 

95  7695  7790  7885  7980  8075  8170  8265  8360  8455  8550  95 

96  7776  7872  7968  8064  8160  8256  8352  8448  8544  8640  96 

97  7857  7954  8051  8148  8245  8342  8439  8536  8633  8730  97 

98  7938  8036  8134  8232  8330  8428  8526  8624  8722  8820  98 

99  8019  8118  8217  8316  8415  8514  8613  8712  8811  8910  99 
100  8100  8200  8300  8400  8500  8600  8700  9800  8900  9000  100 

81  82  83  84  85   86  87  88  89  90 


188  MENTAL   AND  SOCIAL  MEASUREMENTS. 

91   92   93   94   95   96   97   98   99   100 


1 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

1 

2 

182 

184 

186 

188 

190 

192 

194 

196 

198 

200 

2 

3 

273 

276 

279 

282 

285 

288 

291 

294 

297 

300 

3 

4 

304 

368 

372 

376 

380 

384 

388 

392 

396 

400 

4 

5 

455 

460 

465 

470 

475 

480 

485 

490 

495 

500 

5 

6 

546 

552 

558 

564 

570 

576 

582 

588 

594 

600 

6 

7 

637 

644 

651 

658 

665 

672 

679 

686 

693 

700 

7 

8 

728 

736 

744 

752 

760 

768 

776 

784 

792 

800 

8 

9 

819 

828 

837 

846 

855 

864 

873 

882 

891 

900 

9 

.0 

910 

920 

930 

940 

950 

960 

970 

980 

990 

1000 

10 

11  1001  1012  1023  1034  1045  1056  1067  1078  1089  1100  11 

12  1092  1104  1116  1128  1140  1152  1164  1176  1188  1200  12 

13  1183  1196  1209  1222  1235  1248  1261  1274  1287  1300  13 

14  1274  1288  1302  1316  1330  1344  1358  1372  1386  1400  14 

15  1365  1380  1395  1410  1425  1440  1455  1470  1485  1500  15 

16  1456  1472  1488  1504  1520  1536  1552  1568  1584  1600  16 

17  1547  1564  1581  1598  1615  1632  1649  1666  1683  1700  17 

18  1638  1656  1674  1692  1710  1728  1746  1764  1782  1800  18 

19  1729  1748  1767  1786  1805  1824  1843  1862  1881  1900  19 

20  1820  1840  1860  1880  1900  1920  1940  1960  1980  2000  20 

21  1911  1932  1953  1974  1995  2016  2037  2058  2079  2100  21 

22  2002  2024  2046  2068  2090  2112  2134  2156  2178  2200  22 

23  2093  2116  2139  2162  2185  2208  2231  2254  2277  2300  23 

24  2184  2208  2232  2256  2280  2304  2328  2352  2376  2400  24 

25  .   2275  2300  2325  2350  2375  2400  2425  2450  2475  2500  25 

26  2366  2392  2418  2444  2470  2496  2522  2548  2574  2600  26 

27  2457  2484  2511  2538  2565  2592  2619  2646  2673  2700  27 

28  2548  2576  2604  2632  2660  2688  2716  2744  2772  2800  28 

29  2639  2668  2fi97  2726  2755  2784  2813  2842  2871  2900  29 

30  2730  2760  2790  2820  2850  2880  2910  2940  2970  3000  30 

31  2821  2852  2883  2914  2945  2976  3007  3038  3069  3100  31 

32  2912  2944  2976  3'»08  3040  3072  3104  3136  3168  3200  32 

33  3003  3036  3069  3102  3135  3168  3201  3234  3267  3300  33 

34  3094  3128  3162  3196  3230  3264  3298  3332  3366  3400  34 

35  3185  3220  3255  3290  3325  3360  3395  3430  3465  3500  35 

36  3276  3312  3348  3384  3420  3456  3492  3528  3564  3600  36 

37  3367  3404  3441  3478  3515  3552  3589  3626  3663  3700  37 

38  3458  3496  3534  3572  3610  3648  3686  3724  3762  3800  38 

39  3549  3588  3627  3666  3705  3744  3783  3822  3861  3900  39 

40  3640  3680  3720  3760  3800  3840  3880  3920  3960  4000  40 

41  3731  3772  3813  3854  3895  3936  3977  4018  4059  4100  41 

42  3822  38H4  3906  3948  3990  4032  4074  4116  4158  4200  42 

43  3913  3956  3999  4042  4085  4128  4171  4214  4257  4300  43 

44  4oo4  4048  4092  4136  4180  4224  4268  4312  4356  4400  44 

45  4095  4140  4185  4230  4275  4320  4365  4410  4455  4500  45 

46  4186  4232  4278  4324  4370  4416  4462  4508  4554  4600  46 

47  4277  4324  4371  4418  4465  4512  4559  4606  4653  4700  47 

48  4368  4416  4464  4512  4560  4608  4656  4704  4752  4800  48 

49  4459  4508  4557  4606  4655  4704  4753  4802  4851  4900  49 

50  4550  4600  4650  4700  4750  4800  4850  4900  4950  5000  50 

91  92  93   94  95  96  97   98   99  100 


A  MULTIPLICATION  TABLE.  189 

91   92   93  94   95  96  97  98  99  100 

51  4641  4692  4743  4794  4845  4896  4947  4998  5049  5100  51 

52  4732  4784  4836  4888  4940  4992  5044  5096  5148  5200  52 

53  4823  4876  4929  4982  5035  5088  5141  5194  5247  5300  53 

54  4914  4968  5022  5076  5130  5184  5238  5292  5346  5400  54 

55  5005  5060  5115  5170  5225  5280  5335  5390  5445  5500  55 

56  5096  5152  5208  5264  5320  5376  5432  5488  5544  5600  56 

57  5187  5244  5301  5358  5415  5472  5529  5536  5643  5700  57 

58  5278  5336  5394  5452  5510  5568  5626  5684  5742  5800  58 

59  5369  5428  5487  5546  5605  5664  5723  5782  5841  5900  59 

60  5460  5520  5580  5640  5700  5760  5820  5880  5940  6000  60 

61  5551  5612  5673  5734  5795  5856  5917  5978  6039  6100  61 

62  5642  5704  5766  5828  5890  5952  6014  6076  6138  6200  62 

63  5733  5796  5859  5922  5985  6048  6111  6174  6237  6300  63 

64  5824  5888  5952  6016  6080  6144  6208  6272  6336  6400  64 

65  5915  5980  6045  6110  6175  6240  6305  6370  6435  6500  65 

66  6006  6072  6138  6204  6270  6336  6402  6468  6534  6600  66 

67  6097  6164  6231  6298  6365  6432  6499  6566  6633  6700  67 

68  6188  6256  6324  6392  6460  6528  6596  6664  6732  6800  68 

69  6279  6348  6417  6486  6555  6624  6693  6762  6831  6900  69 

70  6370  6440  6510  6580  6650  6720  6790  6860  6930  7000  70 

71  6461  6532  6603  6674  6745  6816  6887  6958  7029  7100  71 

72  6552  6624  6696  6768  6840  6912  6984  7056  7128  7200  72 

73  6643  6716  6789  6862  6935  7008  7081  7154  7227  7300  73 

74  6734  6808  6882  6956  7030  7104  7178  7252  7326  7400  74 

75  6825  6900  6975  7050  7125  7200  7275  7350  7425  7500  75 

76  6916  6992  7068  7144  7220  7296  7372  7448  7524  7600  76 

77  7007  7084  7161  7238  7315  7392  7469  7546  7623  7700  77 

78  7098  7176  7254  7332  7410  7488  7566  7644  7722  7800  78 

79  7189  7268  7347  7426  7505  7584  7663  7742  7821  7900  79 

80  7280  7360  7440  7520  7600  7680  7760  7840  7920  8000  80 

81  7371  7452  7533  7614  7695  7776  7857  7938  8019  8100  81 

82  7462  7544  7626  7708  7790  7872  7954  8036  8118  8200  82 

83  7553  7636  7719  7802  7885  7968  8051  8134  8217  8300  83 

84  7644  7728  7812  7896  7980  8064  8148  8232  8316  8400  84 

85  7735  7820  7905  7990  8075  8160  8245  8330  8415  8500  85 

86  7826  7912  7998  8084  8170  8256  8342  8428  8514  8600  86 

87  7917  8004  8091  8178  8265  8352  8439  8526  8613  8700  87 

88  8008  8096  8184  8272  8360  8448  8536  8624  8712  8800  88 

89  8099  8188  8277  8366  8455  8544  8633  8722  8811  8900  89 

90  8190  8280  8370  8460  8550  8640  8730  8820  8910  9000  90 

91  8281  8372  8463  8554  8645  8736  8827  8918  9009  9100  91 

92  8372  8464  8556  8648  8740  8832  8924  9016  9HM  9200  92 

93  8463  8556  8649  8742  8835  8928  9021  9114  9207  9300  93 

94  8554  8648  8742  8836  8930  9024  9118  9212  9306  9  loo  94 

95  8645  8740  8835  8930  9025  9120  9215  9310  9405  9500  95 

96  8736  8832  8928  9024  9120  9216  9312  9408  9504  9600  96 

97  8827  8924  9021  9118  9215  9312  9409  9506  9603  0700  97 

98  8918  9016  9114  9212  9310  9408  9501;  9604  9702  9800  98 

99  9009  9108  9207  9306  9405  9504  9603  9702  980]  oooo  99 
100  9100  9200  9300  9400  9500  9600  0700  9800  9900  L0000  100 

91   92  93  94   95  96  97  98  99  100 


APPENDIX   II. 

A    TABLE    OF    THE    SQUARES    AND    SQUARE    ROOTS    OF    THE 
NUMBERS    FROM    1    TO    1000. 

This  table  is  a  modification  of  the  first  part  of  Barlow's  Tables. 
The  advantage  of  this  abridged  table  beyond  its  more  convenient 
size,  is  that  through  the  omission  of  cubes,  cube  roots  and  reciprocals, 
the  table  allows  more  rapid  use  and  causes  much  less  strain  on  the 
eyes.  The  latter  result  is  furthered  by  giving  square  roots  only  to 
the  third  decimal  instead  of  to  the  seventh. 


190 


TABLE  OF  SQUARES  AND  SQUARE  ROOTS. 


191 


Num. 

Square. 

Squ.  Root 

1 

1 

1.000 

2 

4 

1.414 

3 

9 

1.732 

4 

16 

2.000 

5 

25 

2.236 

6 

36 

2.449 

7 

49 

2.646 

8 

64 

2.828 

9 

81 

3.000 

10 

100 

3.162 

11 

121 

3.317 

12 

144 

3.464 

13 

169 

3.606 

14 

196 

3.742 

15 

2  25 

3.873 

16 

2  56 

4.000 

17 

2  89 

4.123 

18 

3  24 

4.243 

19 

3  61 

4.359 

20 

4  00 

4.472 

21 

4  41 

4.583 

22 

4  84 

4.690 

23 

5  29 

4.796 

24 

5  76 

4.899 

25 

6  25 

5.000 

26 

6  76 

5.099 

27 

7  29 

5.196 

28 

7  84 

5.292 

29 

8  41 

5.385 

30 

9  00 

5.477 

31 

9  61 

5.568 

32 

10  24 

5.657 

33 

10  89 

5.745 

34 

1156 

5.831 

35 

12  25 

5.916 

36 

12  96 

6.000 

37 

13  69 

6.083 

38 

14  44 

6.164 

39 

15  21 

6.245 

40 

16  00 

6.325 

41 

16  81 

6.403 

42 

17  64 

6.481 

43 

18  49 

6.557 

44 

19  36 

6.633 

45 

20  25 

6.708 

46 

21  ]i\ 

6.782 

47 

22  09 

6.856 

48 

23  04 

6.928 

49 

24  01 

7.000 

50 

25  00 

7.071 

Num. 

Square. 

Squ.  Root 

51 

26  01 

7.141 

52 

27  04 

7.211 

53 

28  09 

7.280 

54 

2916 

7.348 

55 

30  25 

7.416 

56 

3136 

7.483 

57 

32  49 

7.550 

58 

33  64 

7.616 

59 

34  81 

7.681 

60 

36  00 

7.746 

61 

37  21 

7.810 

62 

38  44 

7.874 

63 

39  69 

7.937 

64 

40  96 

8.000 

65 

42  25 

8.062 

66 

43  56 

8.124 

67 

44  89 

8.185 

68 

46  24 

8.246 

69 

47  61 

8.307 

70 

49  00 

8.367 

71 

50  41 

8.426 

72 

5184 

8.485 

73 

53  29 

8.544 

74 

54  76 

8.602 

75 

56  25 

8.660 

76 

57  76 

8.718 

77 

59  29 

8.775 

78 

60  84 

8.832 

79 

62  41 

8.888 

80 

64  00 

8.944 

81 

65  61 

9.000 

82 

67  24 

9.055 

83 

68  89 

9.110 

84 

70  56 

9.165 

85 

72  25 

9.220 

86 

73  96 

9.274 

87 

75  69 

9.327 

88 

77  44 

9.381 

89 

79  21 

9.434 

90 

8100 

9.487 

91 

82  81 

9.539 

92 

84  64 

9.592 

93 

86  49 

9.644 

94 

88  36 

9.C.U5 

95 

90  25 

9.747 

96 

92  16 

9.798 

97 

9409 

9.849 

98 

96  04 

9.899 

99 

9801 

9.950 

100 

100  00 

1(1.000 

192 


MENTAL    AND  SOCIAL  MEASUREMENTS. 


Num. 

Square. 

Squ.  Root. 

101 

102  01 

10.050 

102 

104  04 

10.100 

103 

106  09 

10.149 

104 

10816 

10.198 

105 

110  25 

10.247 

106 

112  36 

10.296 

107 

114  49 

10.344 

108 

116  64 

10.392 

109 

118  81 

10.440 

110 

12100 

10.488 

111 

123  21 

10.536 

112 

125  44 

10.583 

113 

127  69 

10.630 

114 

129  96 

10.677 

115 

132  25 

10.724 

116 

134  56 

10.770 

117 

136  89 

10.817 

118 

139  24 

10.863 

119 

14161 

10.909 

120 

144  00 

10.954 

121 

146  41 

11.000 

122 

148  84 

11.045 

123 

15129 

11.091 

124 

153  76 

11.136 

125 

156  25 

11.180 

126 

158  76 

11.225 

127 

16129 

11.269 

128 

163  84 

11.314 

129 

166  41 

11.358 

130 

169  00 

11.402 

131 

17161 

11.446 

132 

174  24 

11.489 

133 

176  89 

11.533 

134 

179  56 

11.576 

135 

182  25 

11.619 

136 

184  96 

11.662 

137 

187  69 

11.705 

138 

190  44 

11.747 

139 

193  21 

11.790 

140 

196  00 

11.832 

141 

198  81 

11.874 

142 

2  0164 

11.916 

143 

2  04  49 

11.958 

144 

2  07  36 

12.000 

145 

2  10  25 

12.042 

146 

21316 

12.083 

147 

216  09 

12.124 

148 

219  04 

12.166 

149 

2  22  01 

12.207 

150 

2  25  00 

12.247 

Sum. 

Square. 

Squ.  Root. 

151 

2  28  01 

12.288 

152 

2  3104 

12.329 

153 

2  34  09 

12.369 

154 

2  3716 

12.410 

155 

2  40  25 

12.450 

156 

2  4336 

12.490 

157 

2  46  49 

12.530 

158 

2  49  64 

12.570 

159 

2  52  81 

12.610 

160 

2  56  00 

12.649 

161 

2  59  21 

12.689 

162 

2  62  44 

12.728 

163 

2  65  69 

12.767 

164 

2  68  96 

12.806 

165 

2  72  25 

12.845 

166 

2  75  56 

12.884 

167 

2  78  89 

12.923 

168 

2  82  24 

12.961 

169 

2  85  61 

13.000 

170 

2  89  00 

13.038 

171 

2  92  41 

13.077 

172 

2  95  84 

13.115 

173 

2  99  29 

13.153 

174 

3  02  76 

13.191 

175 

3  06  25 

13.229 

176 

3  09  76 

13.266 

177 

3  13  29 

13.304 

178 

316  84 

13.342 

179 

3  20  41 

13.379 

180 

3  24  00 

13.416 

181 

3  27  61 

13.454 

182 

3  3124 

13.491 

183 

3  34  89 

13.528 

184 

3  38  56 

13.565 

185 

3  42  25 

13.601 

186 

3  45  96 

13.638 

187 

3  49  69 

13.675 

188 

353  44 

13.711 

189 

3  57  21 

13.748 

190 

3  6100 

13.784 

191 

3  64  81 

13.820 

192 

3  68  64 

13.856 

193 

3  72  49 

13.892 

194 

3  76  36 

13.928 

195 

3  80  25 

13.964 

196 

3  8416 

14.000 

197 

3  88  09 

14.036 

198 

3  92  04 

14.071 

199 

3  96  01 

14.107 

200 

4  00  00 

14.142 

TABLE  OF  SQUARES  AND  SQUARE  ROOTS. 


193 


fnm. 

Square. 

Squ.  Root. 

201 

4  04  01 

14.177 

202 

4  08  04 

14.213 

203 

412  09 

14.248 

204 

41616 

14.283 

205 

4  20  25 

14.318 

206 

4  24  36 

14.353 

207 

4  28  49 

14.387 

208 

4  32  64 

14.422 

209 

4  36  81 

14.457 

210 

4  4100 

14.491 

211 

4  45  21 

14.526 

212 

4  49  44 

14.560 

213 

4  53  69 

14.595 

214 

4  57  96 

14.629 

215 

4  62  25 

14.663 

216 

4  66  56 

14.697 

217 

4  70  89 

14.731 

218 

4  75  24 

14.765 

219 

4  79  61 

14.799 

220 

4  84  00 

14.832 

221 

4  88  41 

14.866 

222 

4  92  84 

14.900 

223 

4  97  29 

14.933 

224 

5  0176 

14.967 

225 

5  06  25 

15.000 

226 

510  76 

15.033 

227 

515  29 

15.067 

228 

519  84 

15.100 

229 

5  24  41 

15.133 

230 

5  29  00 

15.166 

231 

5  33  61 

15.199 

232 

5  38  24 

15.232 

233 

5  42  89 

15.264 

234 

5  47  56 

15.297 

235 

5  52  25 

15.330 

236 

5  56  96 

15.362 

237 

5  6169 

15.395 

238 

5  66  44 

15.427 

239 

5  7121 

15.460 

240 

5  76  00 

15.492 

241 

5  80  81 

15.524 

242 

5  85  64 

15.556 

243 

5  90  49 

15.588 

244 

5  95  36 

15.620 

245 

6  00  25 

15.652 

246 

6  05K; 

15.684 

247 

6  10  09 

15.716 

248 

6  15  04 

15.748 

249 

6  20  01 

15.780 

250 

6  25  00 

15.811 

Num. 

Square. 

Squ.  Root. 

251 

6  30  01 

15.843 

252 

6  35  04 

15.875 

253 

6  40  09 

15.906 

254 

6  45  16 

15.937 

255 

6  50  25 

15.969 

256 

6  55  36 

16.000 

257 

6  60  49 

16.031 

258 

6  65  64 

16.062 

259 

6  70  81 

16.093 

260 

6  76  00 

16.125 

261 

6  8121 

16.155 

262 

6  86  44 

16.186 

263 

6  9169 

16.217 

264 

6  96  96 

16.248 

265 

7  02  25 

16.279 

266 

7  07  56 

16.310 

267 

712  89 

16.340 

268 

718  24 

16.371 

269 

7  23  61 

16.401 

270 

7  29  00 

16.432 

271 

7  34  41 

16.462 

272 

7  39  84 

16.492 

273 

7  45  29 

16.523 

274 

7  50  76 

16.553 

275 

7  56  25 

16.583 

276 

7  6176 

16.613 

277 

7  67  29 

16.643 

278 

7  72  84 

16.673 

279 

7  78  41 

16.703 

280 

7  84  00 

16.733 

281 

7  89  61 

16.763 

282 

7  95  24 

16.793 

283 

8  00  89 

16.823 

284 

8  06  56 

16.852 

285 

812  25 

16.882 

286 

817  96 

16.912 

287 

8  23  69 

16.941 

288 

8  29  44 

16.971 

289 

8  35  21 

17.000 

290 

8  4100 

17.029 

291 

8  46  81 

17.059 

292 

8  52  64 

17.088 

293 

8  58  49 

17.117 

294 

8  64  36 

17.146 

295 

8  70  25 

17.176 

296 

8  76  16 

17.205 

297 

8  82  09 

17.234 

298 

8  88  04 

17.263 

299 

8  94  01 

17.292 

300 

9  00  00 

17.321 

13 


194  MENTAL  AND  SOCIAL  MEASUREMENTS. 


Num. 

Square. 

Squ.  Hoot 

301 

9  06  01 

17.849 

302 

9  12  04 

1 7 .378 

303 

918  09 

17.407 

304 

9  24  10 

17.436 

305 

9  30  25 

17.464 

306 

9  36  36 

17.493 

307 

9  42  49 

17.521 

308 

9  48  64 

17.550 

309 

9  54  81 

17.578 

310 

9  6100 

17.607 

311 

9  67  21 

17.635 

312 

9  73  44 

17.664 

313 

9  79  69 

17.692 

314 

9  85  96 

17  720 

315 

9  92  25 

17.748 

316 

9  98  56 

17.776 

317 

10  04  89 

17.804 

318 

101124 

17.833 

319 

1017  61 

17.861 

320 

10  24  00 

17.889 

321 

10  30  41 

17.916 

322 

10  36  84 

17.944 

323 

10  43  29 

17.972 

324 

10  49  76 

18.000 

325 

10  56  25 

18.028 

326 

10  62  76 

18.055 

327 

10  69  29 

18.083 

328 

10  75  84 

18.111 

329 

10  82  41 

18.138 

330 

10  89  00 

18.166 

331 

10  95  61 

18.193 

332 

11  02  24 

18.221 

333 

11  08  89 

18.248 

334 

1115  56 

18.276 

335 

1122  25 

18  303 

336 

1128  96 

18.330 

337 

11  35  69 

18358 

338 

11  42  44 

18.385 

339 

11  49  21 

18.412 

340 

11  56  00 

18.439 

341 

11  62  81 

18.466 

342 

11  69  64 

18  493 

343 

11  76  49 

18.520 

344 

11  83  36 

18.547 

345 

11  90  25 

18.574 

346 

119716 

18.601 

347 

12  04  09 

18.628 

348 

12  1104 

18.655 

349 

1218  01 

18.682 

350 

12  25  00 

18.708 

Num. 

Square. 

Squ.  Root 

351 

12  32  01 

18.735 

352 

12  39  04 

18.762 

353 

12  46  09 

18.788 

354 

12  5316 

18.815 

355 

12  60  25 

18.841 

356 

12  67  36 

18.868 

357 

12  74  49 

18.894 

358 

12  8164 

18.921 

359 

12  88  81 

18.947 

360 

12  96  00 

18.974 

361 

13  03  21 

19.000 

362 

13  10  44 

19  026 

363 

13  17  69 

19.053 

364 

13  24  96 

19.079 

365 

13  32  25 

19.105 

366 

13  39  56 

19.131 

367 

13  46  89 

19.157 

368 

13  54  24 

19.183 

369 

13  6161 

19.209 

370 

13  69  00 

19  235 

371 

13  76  41 

19.261 

372 

13  83  84 

19.287 

373 

13  9129 

19.313 

374 

13  98  76 

19.339 

375 

14  06  25 

19.365 

376 

1413  76 

19.391 

377 

14  2129 

19.416 

378 

14  28  84 

19.442 

379 

14  36  41 

19  468 

'380 

14  44  00 

19.494 

381 

14  5161 

19.519 

382 

14  59  24 

19.545 

383 

14  66  89 

19.570 

384 

14  74  56 

19.596 

385 

14  82  25 

19  621 

386 

14  89  96 

19.647 

387 

14  97  69 

19.672 

388 

15  05  44 

19.698 

389 

15  13  21 

19.723 

390 

15  21  00 

19.748 

391 

15  28  81 

19.774 

392 

15  36  64 

19.799 

393 

15  44  49 

19.824 

394 

15  52  36 

19  849 

395 

15  60  25 

19.875 

396 

15  6816 

19.900 

397 

15  76  09 

19.925 

398 

15  84  04 

19  950 

399 

15  92  01 

19.975 

400 

16  00  00 

20.000 

TABLE  OF  SQUARES  AND  SQUARE  ROOTS. 


195 


Num. 

Square. 

Squ.  Root 

401 

16  08  01 

20.025 

402 

1616  04 

20.050 

403 

16  24  09 

20.075 

404 

16  32  16 

20.100 

405 

16  40  25 

20.125 

406 

16  48  36 

20.149 

407 

16  56  49 

20.174 

408 

16  64  64 

20.199 

409 

16  72  81 

20.224 

410 

16  8100 

20^248 

411 

16  89  21 

20.273 

412 

16  97  44 

20.298 

413 

17  05  69 

20.322 

414 

1713  96 

20.347 

415 

17  22  25 

20.372 

416 

17  30  56 

20.396 

417 

17  38  89 

20.421 

418 

17  47  24 

20  445 

419 

17  55  61 

20.469 

420 

17  64  00 

20.494 

421 

17  72  41 

20.518 

422 

17  80  84 

20.543 

423 

17  89  29 

20.567 

424 

17  97  76 

20.591 

425 

18  06  25 

20.616 

426 

1814  76 

20.640 

427 

18  23  29 

20.664 

428 

18  31  84 

20.688 

429 

18  40  41 

20.712 

430 

18  49  00 

20.736 

431 

18  57  61 

20.761 

432 

18  66  24 

20.785 

433 

18  74  89 

20.809 

434 

18  83  56 

20.833 

435 

18  92  25 

20.857 

436 

19  00  96 

20  881 

437 

19  09  69 

20.905 

438 

1918  44 

20.928 

439 

19  27  21 

20.952 

440 

19  36  00 

20.976 

441 

19  44  81 

21.000 

442 

19  5:;  64 

21.024 

443 

19  62  49 

21.048 

444 

19  71  86 

21.071 

445 

19  80  25 

21.095 

446 

19  89  16 

21.119 

447 

19  98(19 

21.142 

448 

20  07  04 

21.166 

449 

20  16  01 

21.190 

450 

20  25  00 

21.213 

Num. 

Square. 

Squ.  Root 

451 

20  34  01 

21.237 

452 

20  43  04 

21.260 

453 

20  52  09 

21.284 

454 

20  61  16 

21.307 

455 

20  70  25 

21.331 

456 

20  79  36 

21.354 

457 

20  88  49 

21.378 

458 

20  97  64 

21401 

459 

2106  81 

21.424 

460 

2116  00 

21.448 

461 

21  25  21 

21.471 

462 

21  34  44 

21.494 

463 

21  43  69 

21.517 

464 

21  52  96 

21.541 

465 

21  62  25 

21.564 

466 

21  71  56 

21.587 

467 

21  80  89 

21.610 

468 

21  90  24 

21.633 

469 

21  99  61 

21.656 

470 

22  09  00 

21.679 

471 

22  18  41 

21.703 

472 

22  27  84 

21.726 

473 

22  37  29 

21.749 

474 

22  46  76 

21.772 

475 

22  56  25 

21.794 

476 

22  65  76 

21.817 

477 

22  75  29 

21.840 

478 

22  84  84 

21.863 

479 

22  94  41 

21.886 

480 

23  04  00 

21.909 

481 

2313  61 

21.932 

482 

23  23  24 

21.954 

483 

23  32  89 

21.977 

484 

23  42  56 

22.000 

485 

23  52  25 

22.023 

486 

23  61  96 

22  045 

487 

23  71  69 

22.068 

488 

23  8144 

22.091 

489 

23  91  21 

22.113 

490 

24  01  00 

22.186 

491 

2410  81 

22.159 

492 

2420  64 

22.181 

498 

24  30  49 

22.204 

494 

244086 

22.226 

495 

24  50  25 

22.249 

496 

2460  16 

22.271 

497 

2170  09 

22  298 

498 

2  1  so  01 

22^816 

499 

249001 

22  888 

500 

25  0000 

22  861 

196 


MENTAL   AND  SOCIAL  MEASUREMENTS. 


Num. 

Square. 

Si|U.  Root. 

Num. 

Square. 

Squ.  Root. 

501 

25  10  01 

22.383 

551 

30  36  01 

23.473 

502 

25  20  04 

22.405 

552 

30  47  04 

23.495 

503 

25  30  09 

22.428 

553 

30  58  09 

23.516 

504 

25  40  16 

22.450 

554 

30  69  16 

23.537 

505 

25  50  25 

22.472 

555 

30  80  25 

23.558 

506 

25  60  36 

22.494 

556 

30  91  36 

23.580 

507 

25  70  49 

22.517 

557 

31  02  49 

23.601 

508 

25  80  64 

22.539 

558 

3113  64 

23.622 

509 

25  90  81 

22.561 

559 

31  24  81 

23.643 

510 

26  01  00 

22.583 

560 

31  36  00 

23.664 

511 

261121 

22.605 

561 

31  47  21 

23.685 

512 

26  21  44 

22.627 

562 

31  58  44 

23.707 

513 

26  31  69 

22.650 

563 

31  69  69 

23.728 

514 

26  41  96 

22.672 

564 

31  80  96 

23.749 

515 

26  52  25 

22.694 

565 

31  92  25 

23.770 

516 

26  62  56 

22.716 

566 

32  03  56 

23.791 

517 

26  72  89 

22.738 

567 

32  14  89 

23.812 

518 

26  83  24 

22.760 

568 

32  26  24 

23.833 

519 

26  93  61 

22.782 

569 

32  37  61 

23.854 

520 

27  04  00 

22.804 

570 

32  49  00 

23.875 

521 

27  14  41 

22.825 

571 

32  60  41 

23  896 

522 

27  24  84 

22.847 

572 

32  71  84 

23.917 

523 

27  35  29 

22.869 

573 

32  83  29 

23.937 

524 

27  45  76 

22.891 

574 

32  94  76 

23.958 

525 

27  56  25 

22.913 

575 

33  06  25 

23.979 

526 

27  66  76 

22.935 

576 

33  17  76 

24.000 

527 

27  77  29 

22.956 

577 

33  29  29 

24.021 

528 

27  87  84 

22.978 

578 

33  40  84 

24.042 

529 

27  98  41 

23.000 

579 

33  52  41 

24.062 

530 

28  09  00 

23.022 

580 

33  64  00 

24.083 

531 

28  19  61 

23.043 

581 

33  75  61 

24.104 

532 

28  30  24 

23.065 

582 

33  87  24 

24.125 

533 

28  40  89 

23.087 

583 

33  98  89 

24.145 

534 

28  5156 

23.108 

584 

34  10  56 

24.166 

535 

28  62  25 

23.130 

585 

34  22  25 

24.187 

536 

28  72  96 

23.152 

586 

34  33  96 

24.207 

537 

28  83  69 

23.173 

587 

34  45  69 

24.228 

538 

28  94  44 

23.195 

528 

34  57  44 

24.249 

539 

29  05  21 

23.216 

589 

34  69  21 

24.269 

540 

29  16  00 

23.238 

590 

34  81  00 

24.290 

541 

29  26  81 

23.259 

591 

34  92  81 

24.310 

542 

29  37  64 

23.281 

592 

35  04  64 

24.331 

543 

29  48  49 

23.302 

593 

35  16  49 

24.352 

544 

29  59  36 

23.324 

594 

35  28  36 

24.372 

545 

29  70  25 

23.345 

595 

35  40  25 

24.393 

546 

29  8116 

23.367 

596 

35  52  16 

24.413 

547 

29  92  09 

23.388 

597 

35  64  09 

24.434 

548 

30  03  04 

23.409 

598 

35  76  04 

24.454 

549 

3014  01 

23.431 

599 

35  88  01 

24.474 

550 

30  25  00 

23.452 

600 

36  00  00 

24.495 

TABLE  OF  SQUARES  AND  SQUARE  ROOTS.  197 


Num. 

Square. 

Squ.  Root. 

Num. 

Square. 

Squ.  Root 

601 

36  12  01 

24.515 

651 

42  38  01 

25.515 

602 

36  24  04 

24.536 

652 

42  51  04 

25.534 

603 

36  36  09 

24.556 

653 

42  64  09 

25.554 

604 

36  48  16 

24.576 

654 

42  77  16 

25.573 

605 

36  60  25 

24.597 

655 

42  90  25 

25.593 

606 

36  72  36 

24  617 

656 

43  03  36 

25.612 

607 

36  84  49 

24.637 

657 

4316  49 

25.632 

608 

36  96  64 

24.658 

658 

43  29  64 

25.652 

609 

37  08  81 

24.678 

659 

43  42  81 

25.671 

610 

37  2100 

24.698 

660 

43  56  00 

25.690 

611 

37  33  21 

24.718 

661 

43  69  21 

25.710 

612 

37  45  44 

24.739 

662 

43  82  44 

25.729 

613 

37  57  69 

24.759 

663 

43  95  69 

25.749 

614 

37  69  96 

24.779 

664 

44  08  96 

25.768 

615 

37  82  25 

24.799 

665 

44  22  25 

25.788 

616 

37  94  56 

24.819 

666 

44  35  56 

25.807 

617 

38  06  89 

24.839 

667 

44  48  89 

25.826 

618 

38  19  24 

24.860 

668 

44  62  24 

25.846 

619 

38  31  61 

24.880 

669 

44  75  61 

25.865 

620 

38  44  00 

24.900 

670 

44  89  00 

25.884 

621 

38  56  41 

24.920 

671 

45  02  41 

25.904 

622 

38  68  84 

24.940 

672 

45  15  84 

25. '.12:'. 

623 

38  81  29 

24.960 

673 

45  29  29 

25.942 

624 

38  93  76 

24.980 

674 

45  42  76 

25.962 

625 

39  06  25 

25.000 

675 

45  56  25 

25.981 

626 

39  18  76 

25.020 

676 

45  69  76 

26.000 

627 

39  3129 

25.040 

677 

45  83  29 

26.019 

628 

39  43  84 

25.060 

678 

45  96  84 

26  038 

629 

39  56  41 

25.080 

679 

46  10  41 

36.058 

630 

39  69  00 

25.100 

680 

46  24  00 

26.077 

631 

39  81  61 

25.120 

681 

46  37  61 

26.096 

632 

39  94  24 

25.140 

682 

46  51  24 

2(1.115 

633 

40  06  89 

25.159 

683 

46  64  89 

26.  KM 

634 

40  19  56 

35.179 

684 

46  78  56 

26.153 

635 

40  32  25 

25  199 

685 

46  92  2.1 

26.173 

636 

40  44  96 

25.219 

686 

47  05  96 

26.192 

637 

405769 

25.239 

687 

4719  69 

26.211 

638 

40  70  44 

25.259 

ess 

47  38  44 

26  280 

639 

4083  21 

25.278 

689 

47  47  21 

26.249 

640 

40  96  00 

25.298 

690 

47  61  00 

26.268 

641 

4108  81 

25.318 

691 

47  74  81 

26.287 

642 

41  21  64 

25.388 

692 

47  8S6I 

26.306 

643 

41  34  49 

25.357 

693 

48  02  1!) 

26.825 

644 

41  47  36 

25  3  7 

694 

is  L6  36 

2i;.:;  u 

646 

41  60  25 

25.3  7 

695 

48  31)2.-. 

26.868 

646 

417:'.  16 

25.417 

696 

48  41  16 

26.882 

647 

41  86  0!) 

25.486 

697 

48  58  09 

26.401 

648 

41  99  04 

85.456 

698 

48  72(»l 

26.420 

6  19 

42  12  01 

25.475 

699 

■ISSCOI 

26.489 

650 

42  25  00 

25. 4!  15 

700 

19  0000 

26.458 

198 


MENTAL    AXD  SOCIAL  MEASUREMENTS. 


Num. 

Square. 

Squ.  Root. 

Num. 

Square. 

Squ.  Root. 

701 

49  14  01 

26.470 

751 

56  40  01 

27  404 

702 

492804 

26.495 

752 

56  55  04 

27.423 

703 

1H42  09 

26.514 

753 

56  70  09 

27.441 

704 

4!)  56  16 

26.533 

754 

56  85  16 

27.459 

705 

49  70  25 

26.552 

755 

57  00  25 

27.477 

706 

49  84  36 

26.571 

756 

57  15  36 

27.495 

707 

49  98  49 

26.589 

757 

57  30  49 

27.514 

708 

50  12  64 

26.608 

758 

57  45  64 

27.532 

709 

50  26  81 

26  627 

759 

57  60  81 

27.550 

710 

50  41  00 

26.646 

760 

57  76  00 

27.568 

711 

50  55  21 

26.665 

761 

57  9121 

27.586 

712 

50  69  44 

26.683 

762 

58  06  44 

27.604 

713 

50  83  69 

26.702 

763 

58  21  69 

27.622 

714 

50  97  96 

26  721 

764 

58  36  96 

27.641 

715 

5112  25 

26.739 

765 

58  52  25 

27.659 

716 

51  26  56 

26.758 

766 

58  67  56 

27.677 

717 

51  40  89 

26.777 

767 

58  82  89 

27.695 

718 

51  55  24 

26.796 

768 

58  98  24 

27.713 

719 

51  69  61 

26.814 

769 

59  13  61 

27.731 

720 

51  84  00 

26.833 

770 

59  29  00 

27.749 

721 

51  98  41 

26.851 

771 

59  44  41 

27.767 

722 

52  12  84 

26.870 

772 

59  59  84 

27.785 

723 

52  27  29 

26.889 

773 

59  75  29 

27.803 

724 

52  41  76 

26.907 

774 

59  90  76 

27.821 

725 

52  56  25 

26.926 

775 

60  06  25 

27.839 

726 

52  70  76 

26.944 

776 

60  21  76 

27.857 

727 

52  85  29 

26.963 

777 

60  37  29 

27.875 

728 

52  99  84 

'26.981 

778 

60  52  84 

27.893 

729 

5314  41 

27  000 

779 

60  68  41 

27.911 

730 

53  29  00 

27.019 

780 

60  84  00 

27.928 

731 

53  43  61 

27.037 

781 

60  99  61 

27.946 

732 

53  58  24 

27.055 

782 

61  15  24 

27.964 

733 

53  72  89 

27.074 

783 

6130  89 

27.982 

734 

53  87  56 

27.092 

784 

61  46  56 

28.000 

735 

54  02  25 

27.111 

785 

61  62  25 

28.018 

736 

5416  96 

27.129 

786 

61  77  96 

28.036 

737 

54  31  69 

27.148 

787 

61  93  69 

28.054 

738 

54  46  44 

27.166 

788 

62  09  44 

28.071 

739 

54  6121 

27.185 

789 

62  25  21 

28.089 

740 

54  76  00 

27.203 

790 

62  41  00 

28.107 

741 

54  90  81 

27.221 

791 

62  56  81 

28.125 

742 

55  05  64 

27.240 

792 

62  72  64 

28.142 

743 

55  20  49 

27.258 

793 

62  88  49 

28.160 

744 

55  35  36 

27.276 

794 

63  04  36 

28.178 

745 

55  50  25 

27.295 

795 

63  20  25 

28.196 

746 

55  6516 

27.313 

796 

63  36  16 

28.213 

747 

55  80  09 

27.331 

797 

63  52  09 

28.231 

748 

55  95  04 

27.350 

798 

63  68  04 

28.249 

749 

56  10  01 

27.368 

799 

63  84  01 

28.267 

750 

56  25  00 

27.386 

800 

64  00  00 

28.284 

TABLE  OF  SQUARES  AND  SQUARE  ROOTS. 


199 


Num. 

Square. 

Squ.  Root. 

801 

64  16  01 

28.302 

802 

64  32  04 

28.320 

803 

64  48  09 

28.337 

804 

64  64  16 

28.355 

805 

64  80  25 

28.373 

806 

64  96  36 

28.390 

807 

65  12  49 

28.408 

808 

65  28  64 

28.425 

809 

65  44  81 

28.443 

810 

65  61  00 

28.460 

811 

65  77  21 

28.478 

812 

65  93  44 

28.496 

813 

66  09  69 

28.513 

814 

66  25  96 

28.531 

815 

66  42  25 

28.548 

816 

66  58  56 

28  566 

817 

66  74  89 

28.583 

818 

66  91  24 

28.601 

819 

67  07  61 

28.618 

820 

67  24  00 

28.636 

821 

67  40  41 

28.653 

822 

67  56  84 

28.671 

823 

67  73  29 

28.688 

824 

67  89  76 

28.705 

825 

68  06  25 

28.723 

82S 

68  22  76 

28.740 

827 

68  39  29 

28.758 

828 

68  55  84 

28.775 

829 

68  72  41 

28.792 

830 

68  89  00 

28.810 

831 

69  05  61 

28.827 

832 

69  22  24 

28.844 

833 

69  38  89 

28.862 

834 

69  55  56 

28.879 

835 

69  72  25 

28.896 

836 

69  88  96 

28.914 

837 

70  05  69 

28.931 

838 

70  22  44 

28  948 

839 

70  39  21 

28.965 

840 

70  56  00 

28.983 

841 

70  72  81 

29.000 

842 

70  89  64 

29.017 

843 

71  06  49 

29.034 

844 

71  23  36 

29  052 

845 

71  40  25 

29.009 

846 

71  57  16 

29.086 

847 

71  74  09 

29.103 

848 

71  91  04 

29.120. 

849 

72  08  01 

29.138 

850 

72  25(H) 

29.155 

Num. 

Square. 

Squ.  Root. 

851 

72  42  01 

29.172 

852 

72  59  04 

29.189 

853 

72  76  09 

29.206 

854 

72  93  16 

29.223 

855 

73 10  25 

29.240 

856 

73  27  36 

29.257 

857 

73  44  49 

29.275 

858 

73  61  64 

29.292 

859 

73  78  81 

29.309 

860 

73  96  00 

29.326 

861 

74  13  21 

29.343 

862 

74  30  44 

29  360 

863 

74  47  69 

29.377 

864 

74  64  96 

29.394 

865 

74  82  25 

29.411 

866 

74  99  56 

29.428 

867 

75  16  89 

29.445 

868 

75  34  24 

29.462 

869 

75  51  61 

29.479 

870 

75  69  00 

29.496 

871 

75  86  41 

29.513 

872 

76  03  84 

29.530 

873 

76  21  29 

29.547 

874 

76  38  76 

29.563 

875 

76  56  25 

29.580 

876 

76  73  76 

29.597 

877 

76  91  29 

29.614 

878 

77  08  84 

29.631 

879 

77  26  41 

29.648 

880 

77  44  00 

29.665 

881 

77  61  61 

29.682 

882 

77  79  24 

29.698 

883 

77  96  89 

29.715 

884 

78  14  56 

29.732 

885 

78  32  25 

29.749 

886 

78  49  96 

29.766 

887 

78  67  69 

29.783 

888 

78  85  44 

29.799 

889 

79  03  21 

29.816 

890 

79  21  00 

29.833 

891 

79  38  81 

29.850 

892 

79  56  64 

29.866 

893 

79  74  49 

29.883 

894 

79  92  36 

29.900 

895 

80 10  25 

29.916 

896 

80  28  16 

29.933 

897 

80  4C  (>'.» 

29.950 

898 

80  64  04 

29.967 

899 

80  82  01 

29.983 

900 

81  00  00 

30.000 

200 


MENTAL  AND  SOCIAL  MEASUREMENTS. 


Num. 

Square. 

Sqll.  Root. 

Num. 

Square. 

Squ.  Root. 

901 

81 18  01 

30.017 

951 

90  44  01 

30.838 

902 

81  36  04 

30.033 

952 

90  63  04 

30.854 

903 

81  .->4  09 

30.050 

958 

90  82  09 

30.871 

904 

81  72  16 

30.067 

954 

91  01 16 

30.887 

905 

81  90  25 

30.083 

955 

91  20  25 

30.903 

906 

82  08  36 

30.100 

956 

91  39  36 

30.919 

907 

82  26  49 

30.116 

957 

91  58  49 

30.935 

908 

82  44  64 

30.133 

958 

91  77  64 

30.952 

909 

82  62  81 

30.150 

959 

9196  81 

30.968 

910 

82  81  00 

30.166 

960 

92 16  00 

30.984 

911 

82  99  21 

30.183 

961 

92  35  2L 

31.000 

912 

83  17  44 

30.199 

962 

92  54  44 

31.016 

913 

83  35  69 

30.216 

963 

92  73  69 

31.032 

914 

83  53  96 

30.232 

964 

92  92  96 

31.048 

915 

83  72  25 

30.249 

965 

93  12  25 

31.064 

916 

83  90  56 

30.265 

966 

93  31  56 

31.081 

917 

84  08  89 

30.282 

967 

93  50  89 

31.097 

918 

84  27  24 

30.299 

968 

93  70  24 

31.113 

919 

84  45  61 

30.315 

969 

93  89  61 

31.129 

920 

84  6400 

30.332 

970 

94  09  00 

31.145 

921 

84  82  41 

30.348 

971 

94  28  41 

31.161 

922 

85  00  84 

30.364 

972 

94  47  84 

31.177 

923 

85  19  29 

30.381 

973 

94  67  29 

31.193 

924 

85  37  76 

30.397 

974 

94  86  76 

31.209 

925 

85  56  25 

30.414 

975 

95  06  25 

31.225 

926 

85  74  76 

30.430 

976 

95  25  76 

31.241 

927 

85  93  29 

30.447 

977 

95  45  29 

31.257 

928 

86  11  84 

30.463 

978 

95  64  84 

31.273 

929 

86  30  41 

30.480 

979 

95  84  41 

31.289 

930 

86  49  00 

30.496 

980 

96  04  00 

31.305 

931 

86  67  61 

30.512 

981 

96  23  61 

31.321 

932 

86  86  24 

30.529 

982 

96  43  24 

31.337 

933 

87  04  89 

30.545 

983 

96  62  89 

31.353 

934 

87  23  56 

30.561 

984 

96  82  56 

31.369 

935 

87  42  25 

30.578 

985 

97  02  25 

31.385 

936 

87  60  96 

30.594 

986 

97  2196 

31.401 

937 

87  79  69 

30.610 

987 

97  41  69 

31.417 

938 

87  98  44 

30.627 

988 

97  61  44 

31.432 

939 

88  17  21 

30.643 

989 

97  81  21 

31.448 

940 

88  36  00 

30.659 

990 ' 

98  01  00 

31.464 

941 

88  54  81 

30.676 

991 

98  20  81 

31.480 

942 

88  73  64 

30.692 

992 

98  40  64 

31.496 

943 

88  92  49 

30.708 

993 

98  60  49 

31.512 

944 

89  11  36 

30.725 

994 

98  80  36 

31.528 

945 

89  30  25 

30.741 

995 

99  00  25 

31.544 

946 

89  49  16 

30.757 

996 

99  2016 

31.559 

947 

89  68  09 

30.773 

997 

99  40  09 

31.575 

948 

89  87  04 

30.790 

998 

99  60  04 

31.591 

949 

90  06  01 

30.806 

999 

99  80  01 

31.607 

950 

90  25  00 

30.822 

1000 

100  00  00 

31.623 

APPENDIX   III. 

ANSWERS    TO    PROBLEMS;     MISCELLANEOUS    PROBLEMS. 

Answers  to  Problems. 

7.  $1,312,  since  salaries  between  1,000  and  1,100  are  to  be  reck- 
oned as  averaging  1,050,  and  similarly  for  the  other  groups. 


16. 

Average. 

A.  D. 

a. 

Median. 

25  percentile. 

75  percenti 

I. 

15.0 

1.0 

1.5 

15.1 

14.3 

16.0 

II. 

163.6 

5.8 

7.35 

164.0 

159.1 

168.3 

III. 

14.1 

3.4 

4.3 

13.8 

11.2 

16.7 

17.  Case  I.  Av.  =  40.6.  A.  D.,  a  and  P.  E.  from  Av.  =  re- 
spectively 7.5,  9.6  and  6.4.  Median  =  40.2.  A.  D.,  a  and  P.  E. 
from  median  =  respectively  7.5,  9.6  and  6.4. 

Case  II.  Av.  =  98.58.  A.  D.,  a  and  P.  E.  from  Av.  =  re- 
spectively .51,  .68  and  .41.  Median  =  98.61.  A.  D.,  a  and  P.  E. 
from  median  =  respectively  .51,  .68  and  .41. 

18.  The  great  frequency  of  measures  98.0,  99.0  and  98.6  is  prob- 
ably due  to  the  tendency  of  the  observer  to  record  even  numbers  and 
the  ' normal'  temperature.  The  two  cases  reported  96.0  Avere  very 
likely  observed  simply  as  between  96  and  97  and  then  by  an  error 
recorded  as  96.0.     Av.  =  98.58.     A.  D.  =  .53. 

19.  Case  I.  The  average  is  155.6  ;  the  A.  D.  from  it  of  the  cases 
above  it  is  18  ;  that  of  the  cases  below  it  is  15.  50  per  cent,  of 
the  cases  above  it  deviate  less  than  12.9  from  it.  50  per  cent,  of 
the  cases  below  it  deviate  less  than  13.3  from  it.  75  per  cent,  of  the 
cases  above  it  deviate  less  than  25.2  from  it.  75  per  cent,  of  the  cases 
below  it  deviate  less  than  22.1  from  it.  The  mode  is  the  140-149 
group.  Using  145  as  an  approximate  modal  point,  the  A.  D.  from 
the  mode  of  the  cases  above  it  is  20.8  ;  that  of  those  below  it  is  1 1 .0. 
50  per  cent,  of  the  cases  above  it  deviate  less  than  17.0  from  it.  50 
per  cent,  of  all  the  cases  below  it  deviate  less  than  !).!)  from  it.  75 
per  cent,  of  the  cases  above  it  deviate  less  than  29.6  from  it.  75  per 
cent,  of  the  cases  below  it  deviate  less  than  17.1  from  it. 

19.  Case  II.  The  average  is  5.24;  the  A.  I),  from  it  of  the 
cases  above  it  is  1.2;  that  of  those  below  it   is  .5.     (i()  per  cent. 

201 


202 


VESTAL   AND  SOCIAL   MEASUREMENTS. 


of  the  cases  above  deviate  less  than  1.0  from  the  average.  53.5  per 
cent,  of  the  cases  below  deviate  less  than  .5  from  the  average.  The 
mode  is  5.000 ;  the  A.  D.  from  it  of  the  cases  above  it  is  1.43  ;  that 
of  those  below  it  is  .51.  61  per  cent,  of  the  cases  above  it  deviate 
less  than  1.25.  94.5  per  cent,  of  the  cases  below  it  deviate  less 
than  .50. 

20.  The  mode  and  median  and  P.  E.'s  from  them  and  various 
percentile  values. 

21.  If  the  form  of  distribution  is  a  rectangle, 

A  =  +  1.96  A.  D.  D  =  -  1.22  A.  D. 

B  =  +  1.48  A.  D.  E=-  1.84  A.  D. 

C=  +    .16  A.  D.  F=  -  1.98  A.  D. 

If  the  form  of  distribution  is  that  of  the  normal  probability 
surface, 

4  =  +  3.1A.D.  D  =  -  1.1  A.  D. 

B  =  +  1.5  A.  D.  E=  -  2.2  A.  D. 

C=+    .1  A.  D.  .F=-3.4A.  D. 

If  A-B=B-  Cand  B-  C=  C-D,  etc., 
A  =  -f  2.8  A.  D.  or  +  3.2,  according  to  the  correction  made. 
B=  +  1.5  A. 
C=  +  .2  A. 
D=  -  1.1  A. 
£=-2.4A. 
F=-2>.1  A. 

23.  (1)  +  2.2. 
(2)  +  .08. 
(3) +.9. 

24.  Light  blue  -  2;28<r. 
Blue-dark  blue   -  1.00<r. 
Gray -blue  green  —  .08<r. 
Dark  gray-hazel  +  Ala, 

26a.  70  per  cent. 

266.  35  per  cent. 

29.  r  =  +  .48. 

In  the  answers  to  problems  30-42  the  unreliabilities  are  given  in 
terms  of  the  P.  E.true  measure.obtained  measure.  These  can  be  turned  into 
<rt.0  and  A.  D.t  .0  by  multiplying  by  1.4826  and  1.1843  respec- 
tively. 


D. 

"  +  1.7, 

D. 

'+    -2, 

D. 

«  -  1.3, 

D. 

"  -  2.8, 

D.  ' 

'  -  4.3, 

Light  brown-brown 
Dark  brown 


+  .83<t. 
+  1.34<7. 


Very  dark  brown-black  +  2.1 6<r. 


MISCELLANEOUS  PROBLEMS. 


203 


30. 

*  •   -^U.  Av.-obt.  Av.  — 

.22 

,    J.  .   u.t  var.-obt 

var.  =  -16. 

31. 

(I 

.27 

11 

.19. 

32. 

a 

.32; 

a 

.22. 

33. 

a 

.47; 

a 

.34. 

34. 

tt 

.16, 

(( 

.11. 

35. 

" •    -k't.  diff.-obt.  diff.  = 

.39 

36. 

11                        

.52 

37. 

«                         __ 

.27 

38. 

a                    _ 

.31 

39. 

a                  _ 

.36 

40. 

P.  E.,r..oUl,.051. 

41. 

"          .068. 

42. 

"          .039. 

43. 

68.3  per  cent. 

44. 

13.3    "      " 

45. 

.01  "      " 

46. 

11.7    "      " 

47. 

7.9    «      " 

48. 

18.3    "      " 

49. 

4.7    "      " 

50. 

10  and  11.68+. 

51. 

10  and  8.95  +. 

52. 

11.83  +  and  8.07 

— . 

53. 

23.9  and  10.6. 

54. 

9.8  and  the  lower  limit  of  the  distribution  which  will  be 

near  O. 

55. 

9.7  and  22.8. 

56a 

There  are  124  chances  out  of  10,000  for  it. 

566. 

"       "    227 

a            a     {(         u 

a     <( 

56c. 

Between  A.obt  — 

6  and  A.obt  +  6. 

57a. 

There  are  227  chances  out  of  10,000  for  it. 

576. 

"       "  8,664 

a            u     u         a 

(l       u 

58a 

"       "    82 

a             i(     it          (( 

a      u 

586. 

"       "    82 

a            a     a          a 

a      a 

58c. 

"       "  6,828 

<c             ((     a          « 

(t      a 

58c? 

The  chances  are 

20  to  1  that  AY  — 

A  ,  will  exceed  .21  and 

will  not 

exceed  2.19. 

59a. 

Bet  ween..  60  and 

.36. 

204  MENTAL  AND  SOCIAL  MEASUREMENTS. 

596.   227  out  of  10,000. 

60a.   26  out  of  10,000. 

606.   1,160  "  "       " 

61a.  6  out  of  1,000. 

616.  975  "  "     " 

62.  6.73  -f  . 

63a.     60  out  of  1,000. 

636.  190    "    "     " 

63c.    446    "    "     " 

63a7.  212    "    "     « 

63e.    560    "    "     " 

64a.  890  out  of  1,000. 

646.  992    "    "     " 

64c.   19.7  and  12.3. 

65.  As  high  as  .40,  200  chances  in  1,000. 

a        a       a     qQ         q  u  a         a 

67a,  327  out  of  1,000. 
676.  673     "    "      " 

68.  28  per  cent. 

69.  78    "      " 

70.  28    "      " 

MISCELLANEOUS   PROBLEMS. 

71.  Almost  any  statistical  study  of  health  or  crime  or  educational 
work  will  furnish  problems  in  the  selection  of  units  of  measure. 
Amongst  psychological  studies,  those  concerned  with  practice  or 
fatigue  or  changes  due  to  growth  will  be  found  interesting  from  this 
point  of  view. 

72.  Let  the  student  test  himself  with  respect  to  pulse,  strength, 
reaction-time  and  accuracy  of  discrimination  40  times  each,  and  com- 
pute from  the  results  his  central  tendency  and  variability  in  each 
trait.  He  should  guard  against  variations  due  to  the  influence  of 
fatigue  and  practice. 

73.  Record  the  amount  of  sleep  or  exercise  taken  daily  for  a 
month  or  so  and  present  the  facts  in  form  for  statistical  use. 

74.  Calculate  the  median,  the  25  percentile  and  the  75  percentile 
for  each  of  the  traits  measured  in  Tables  VI.  to  XVII. 


MISCELLANEOUS  PROBLEMS.  205 

75.  What  is  the  briefest  expression  of  the  following  facts  that  is 
also  reasonably  adequate  ?  Cost  per  pupil  of  general  school  supplies 
(in  cents)  of  primary  departments  in  Manhattan  and  Bronx  (Report 
of  1901) :  59,  63,  64,  6Q,  67,  68,  69,  70,  73,  75,  75,  75,  76,  77,  77, 
80,  85,  85,  86,  87,  87,  88,  88,  89,  90,  91,  91,  92,  95,  95,  96,  96, 
97,  98,  99,  100,  101,  101,  101,  101,  101,  101,  102,  102,  105,  106, 
107,  109,  110,  110,  110,  113,  117,  118,  120,  122,  123,  124,  124, 
127,  127,  128,  130,  130,  132,  135,  142,  172.  Answer,  median 
98.5  ;  Q.  1.325 ;  distribution  1,  14,  20,  19,  12,  1,  1. 

In  all  examples  that  follow  calculate  the  reliability  of  every 
result  obtained,  whenever  the  data  are  at  hand. 

76.  Calculate  the  central  tendency  and  variability  of  the  follow- 
ing group  measure  : 

Death-Kate  from  Diaeeh(ea  in  Third  Quarter.* 


Quantity. 
0.0 

Frequency. 

1 

Quantity. 
5.0 

Frequency. 
2 

Quantity. 
9.5 

Frequency. 
0 

0.5 

1 

5.5 

0 

10.0 

1 

1.0 

1 

6.0 

8 

10.5 

1 

1.5 

1 

6.5 

3 

11.0 

0 

2.0 

4 

7.0 

3 

11.5 

0 

2.5 

3 

7.5 

2 

12.0 

0 

3.0 

12 

8.0 

3 

12.5 

0 

3.5 

10 

8.5 

1 

13.0 

0 

4.0 
4.5 

7 
5 

9.0 

0 

13.5 

1 

77.  Express  graphically  the  following  group  measure  and  calcu- 
late its  central  tendency  and  variability  : 

Size  of  Schools. f 


Quantity. 

Number  of  Children 

in  the  School. 

Frequency. 

Number 
of  Schools. 

Quantity. 

Number  of  Children 
in  the  School. 

Frequency, 

Number 

ill'  Sri Is. 

Less  than  20 

577 

100-199 

131 

20-29 

821 

200-299 

54 

30-39 

423 

300-399 

38 

40-49 

239 

400-599 

39 

50-99 

252 

600  or  more. 

52 

78.  Calculate  the  central  tendencies  and  variabilities  of  the  two 
group  facts  given  below  and  compare  the  condition  in  I  850  with  that 
in  1891. 

*  From  G.  B.  LangstafT,  Studies  in  Stalixtic.%  p.  299. 
t  From  the  New  South  Wales  Register  of  1901. 


Quantity. 

Per  cent,  of 

Paupers. 

N 

Registr 
In  1850. 

7.5 

44 

8.0 

31 

8.5 

27 

9.0 

34 

9.5 

21 

10.0 

11 

10.5 

12 

11.0 

11 

11.5 

7 

12.0 

7 

12.5 

3 

.    13.0 

1 

13.5 

3 

14.0 

4 

=  588 

632 

206  MENTAL  ASD  SOCIAL  MEASUREMENTS. 

Pauperism  in  England  and  Wales.* 

^T;         SSBTS  *^:,         8H£» 

rer  ceni.  oi  Registration  Districts. 

Paupers.  ini,s.-.0.  In  1891. 

0.5  1 

1.0  4  18 

1.5  2  48 

2.0  7  72 

2.5  11  89 

3.0  21  100 

3.5  28  90 

4.0  33  75 

4.5  46  60 

5.0  55  40 

5.5  40  21 

6.0  45  11 

6.5  44  5 

7.0  35  1 

Total : 

79.  (a)  Present  graphically  the  table  of  frequency  given  below. 
(6)  Present  also  the  distributions  which  would  result  if  selection  so 
worked  on  the  group  that  for  each  removal  of  a  step  from  the  mode 
10  per  cent,  of  the  cases  were  eliminated,  that  is  if  only  90  per  cent, 
of  the  19's  and  21's  remained,  only  80  per  cent,  of  the  18's  and  22's, 
etc.  (c)  Present  also  the  result  if  10  per  cent,  of  the  highest  group 
were  eliminated,  20  per  cent,  of  the  next  highest,  30  per  cent,  of  the 
next,  etc.  (d )  Present  also  the  result  if  there  were  no  elimination 
above  the  mode,  but  below  it  an  elimination  of  3  per  cent,  for  the 
nearest  group,  6  per  cent,  for  the  next,  12  for  the  next,  24  for  the 
next,  etc.  (e)  Let  the  conditions  be  as  in  d  except  that  the  elimina- 
tion be  1  per  cent.,  4,  9,  16,  25,  etc. 

A  Normal  Distribution. 

Quantity.  Frequency.  Quantity.  Frequency. 

11  0.01  21  438 

12  0.2  22  318 

13  2  23  186 

14  7  24  85 

15  29  25  29 

16  85  26  7 

17  186  27  2 

18  318  28  0.2 

19  438  29  0.01 

20  486 

*  From  an  article  by  G.  Udny  Yule  in  the  Journal  of  the  Royal  Statistical  Society, 
Vol.  59,  page  347. 


MISCELLANEOUS  PROBLEMS.  207 

80.  What  is  the  evidence  from  the  figures  themselves  that  the 
form  of  distribution  for  the  rate  of  interest  given  in  the  figures  below 
is  due  to  conventional  rather  than  natural  causes  ? 

Mortgages  on  Homes  in  New  Jersey.* 

Rate  of  Interest.  1 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

ii 

12 
13 
14 
15 
16 
17 
18 
50 

81.  Explain  why  amounts  of  property,  incomes,  holdings  of  land, 
inheritances  and  taxes  should  be  distributed  with  a  mode  at  or  near 
the  low  end  and  a  very  pronounced  positive  skewness,  so  great  that 
the  upper  extreme  is  often  10,000  times  the  amount  of  the  mode. 

82.  Explain  the  form  of  distribution  of  Fig.  87  A. 

83.  The  modal  number  of  children  for  American  women  married 
twenty  years  or  more  was,  in  1700-1750,  seven.  It  is  at  present 
two.  Suppose  a  student  of  the  fertility  of  the  American  race  to  get 
from  a  tabulation  of  the  figures  given  in  genealogy  books  the  distri- 
bution of  Fig.  87  B.     How  would  you  explain  his  result? 

84.  How  would  you  explain  the  form  of  distribution  of  Fig. 
87  C,  found  for  the  frequency  of  pauperism  in  England  and  Wales? 

86.  What  would  be  the  form  of  distribution  of  the  speed  of  race 

horses  ? 

*  Taken  from  a  report  by  G .  H.  Holmes  in  the  Journal  of  the  Royal  statistical 
Society,  Vol.  56,  p.  475. 

t  All  but  160  at  precisely  6  per  cent. 


er  of  Mortgages. 

Amount  of  Mortgages 
(in  thousands  of  dollars). 

104 

136 

5 

6 

15 

36 

64 

130 

387 

790 

10,629 

25,430 

27,956 

38,901t 

428 

407 

69 

68 

30 

33 

57 

58 

8 

7 

23 

18 

2 

1 

0 

0 

6 

3 

0 

less  than  $500. 

1 

less  than  $500. 

1 

less  than  $500. 

1 

20S 


MENTAL    AND  SOCIAL   MEASUREMENTS. 


87.  What  would  be  the  form  of  distribution  of  the  morality  of 
criminals? 

88.  What  would  be  the  form  of  distribution  of  the  intelligence 
of  day-laborers  compared  with  that  of  men  in  general  ? 


12       3        4         5 


10       11       12       13      14 


105  125 


Fig. '87. 

89.  What  would  be  the  form  of  distribution  of  the  weight  of  the 
world's  war- vessels  ? 

90.  Measure  the   difference  between  boys'  and  girls'  grammar 
schools  with  respect  to  the  cost  of  supplies  from  the  following  facts  : 


MISCELLANEOUS  PROBLEMS. 


209 


Cost  per  pupil  of 

general  supplies. 

In  cents. 

Frequ 
Boys' 
Schools. 

ency. 
Girls' 
Schools. 

Cost  per  pupil  of 

general  supplies. 

In  cents. 

Frequency. 

Boys'           Girls' 

Schools.      Schools. 

100  up 

through  109 

1 

250 

5 

3 

110  up 

through  119 

0 

260 

1 

2 

120  etc, 

2 

270 

2 

0 

130 

1 

280 

4 

1 

140 

3 

0 

290 

1 

0 

150 

1 

4 

300 

1 

0 

160 

0 

3 

310 

2 

0 

170 

4 

2 

320 

0 

0 

180 

2 

4 

330 

1 

1 

190 

3 

4 

340 

0 

200 

2 

2 

350 

2 

210 

4 

3 

also 

also 

220 

2 

2 

435 

400 

230 

2 

3 

and 

and 

240 

3 

4 

559 

512 

Answer  :  Gross  difference,  boys'  schools  —  girls'  schools  =  36.6 
cents  if  medians  are  compared.  Only  28  per  cent,  of  girls'  schools 
reach  the  median  mark  for  boys'  schools ;  or  70  per  cent,  of  boys' 
schools  are  more  expensive  than  the  median  girls'  school. 

91.  Compare  the  strength  of  pull  of  men  with  that  of  women, 
using  the  following  facts  :* 


Quantity. 

Frequency  in  Men. 

Frequency  in  Women. 

30 

3 

9 

40 

15 

98 

50 

69 

101 

60 

250 

5 

70 

522 

2 

80 

296 

0 

90 

226 

1 

100 

73 

110 

18 

120 

15 

130 

2 

140 

4 

150 

4 

Total 


1497 


216 


92.  Compare  the  group  of  southern  cities  (vl)  with  the  group  of 
central  western  cities  {B)  with  respect  to  the  regularity  of  attend- 
ance upon  school. 

*From  Appendix  to  C.  Roberts'  Manual  of  Anthropometry. 
14 


210  MENTAL  AND  SOCIAL  MEASUREMENTS. 

Regularity  of  School  Attendance. 

Percentage  Freouencies  Percentage 

attendance  was  rrequeuties.  attendance  was 

of  enrollment.  of  enrollment. 

48  1  74 

50  76 

52  1  78 

54  1  80 

56  1  82 

58  84 

60  1  86 

62  11  88 

64  5      1  90 

66  5      5  92 

68  11      4  94 

70  4      2  96                1 

72  5      3                   63     114 


Frequencies. 
A.                 B. 

9 

10 

4 

18 

8 

23 

2 

20 

1 

12 

1 

6 

2 

3 

2 

1 

1 

1 

\ 


INDEX. 


Accuracy,  of  group  measures,  42  ff.  ;  of 
coefficients  of  correlation,  127  ff. 

Aikins,  H.  A.,  14 

Algebraic  formulation  of  tables  of  fre- 
quency, 34  ff.,  44 

Attenuation  of  coefficients  of  correlation, 
127  ff. 

Averages,  32,  37  f.,  41  ;  reliability  of, 
139  ff.  ;  technique  of  calculating,  71  ff. 

Average  deviation,  33,  38,  59  f.  ;  relia- 
bility of,  142 ;  technique  of  calculat- 
ing, 75 1,  80 

tr,  J.  H.,  46 
',  A.  L.,  2,  47,  70,  162 

i,  methods  of,  see  Technique 
J.  McK.,  19 
and  measurements   of  relation- 
.,,  133  f. 
■al  tendencies,  measurement  of,  32, 
,  37,  38,  41  ;  of  relationships,   117, 
119  f.  ;  reliability  of,  139  ff. 
Changes,  measurement  of,  103  ff.  ;  in  a 
group,  106  ff.  ;  in  an  individual,  104  ff. 
Coefficient  of  correlation,  121  ff.  ;  atten- 
uation of,  127  ff.  ;  reliability  of,  145 
Collet,  Clara,  47 
Constant  errors,  157  ff. 
Constants   determining   a   table   of   fre- 
quency, 32,  34,  44 
Correlation,  110  ff.  ;  attenuation  of,  127 
ff.  ;    Pearson    coefficient    of,    121  ff.  ; 
rectilinear,  119,  122 
Curve  of  error,  36,  59,  60,  69  f . 

Deviations,  see  Variability 

Difference,  measurements  of,  97  ff.,  155 
f.  ;  variability  and,  98  ;  reliability  of, 
142  ff.  ;  zero  points  and,  98 

Distribution,  form  of,  22  ff.,  44  ff.,  52, 
59  f.,  64,  67  f.,  69  f.,  147  ff.  ;  multi- 
modal, 31  f.,  39;  normal,  36,  59  f., 
69  f.,  147  ff.  ;  skewed,  31  f.,  38;  types 
of,  28  ff.,  44  ff. 


Ebbinghaus,  H.,  112,  113. 

Error,  curve  of,  147  ff. 

Errors,  constant,  157  ff.  ;  of  interpreta- 
tion, 159  ff.  ;  of  mean  square,  see 
Standard  deviation ;  sources  of,  157 
ff.  ;  variable,  157  ff. 

Form  of  distribution,  see  Distribution 
Frequency,  surfaces  of,  22  ff. ,  44  ;  tables 
of,  22  ff.,  44 

Gale,  H.  G.,  112 
Galton,  F.,  19,  96 
Grades,  percentile,  38,  79 
Groups,  changes  in,  106  ff.  ;  comparison 
of,  98  ff.,  155  f.  ;  measurement  of,  41  ff. 

Holmes,  G.  H.,  207 
Homogeneity  in  groups,  53  ff. 

Independence  of  causes  and  form  of  dis- 
tribution, 64,  67  f. 

Individuals,  measurement  of,  22  ff. ;  of 
change  in,  104  ff. 

Inheritance,  and  measurements  of  rela- 
tionships, 133 

Langstaff,  G.  B.,  205 
London  Statistics,  104 

Mathematics  and  mental  measurements, 
1  ff. 

Measurements,  by  relative  position,  19 
ff.,  39,  85  ff.,  152  f.;  of  central  tenden- 
cies, 32  ff,,  139  ff. ;  of  changes,  103  ff. ; 
of  differences  97  ff.,  142  ff. ;  of  groups, 
41  ff. ;  of  individuals,  22  ff. ;  of  rela- 
tionships, 110  ff.,  145  ;  of  reliability, 
136  ff.;  of  variability,  38,  59  f.,  75  ff. 

Medians,  37  ;  calculation  of,  74  f.;  re- 
liability of,  140 

Mixture  of  species,  52  ff. 

Modes,  37  ;  calculation  of,  73  f. ;  re- 
liability of,  140 

Multimodality,  31  f.,  39,  52  f.,  54  ff., 
81  f. 


211 


212 


INDEX. 


New  Sovth  Wales  Kegister,  205 
New  Zealand  Official  Year-book, 

84 
Normal  distribution,  36,  59,  60,  69  f., 

147  ft. 

Pearson  coefficient  of  variability,  102  ; 
of  correlation,  121  ff. 

Pearson,  K.,  46,  47,  102 

Percentile  grades,  38,  79  ff. 

Position,  measurement  by  relative,  19 ff., 

39,  85  ff,  101  f.,  152  f. 
Probability,  theory  of,  36,  59  f.,  61  ff., 

137  f.,  143  f. 
Probable  error,  of  a  distribution,  38,  59, 

78  ff.,  142  ;  of  a  measure,  139  ff. 
Problems,  21,   39  f.,  82  ff.,  96,   108  f., 

134 1.,  145  f.,  151,  152,  154,  156,  204 ff. 

Relationships,  measurements  of,  110  ff. ; 
central  tendencies  of,  117,  119,  120  ; 
inheritance  and,  133  ;  reliability  of, 
145  ;  variability  of  110  ff. 

Relative  position,  measurements  by  ;  see 
Position 

Reliability,  of  measures,  136  ff.,  153  f., 
of  central  tendency,  139  ff. ;  of  corre- 
lation, 145  ;  of  difference,  142  ff. ;  of 
variability,  142 

Rice,  J.  M.,  5,  8 

Roberts,  C,  46,  84,  209 

Scales  of  measurements,  15  ff. 
Selection,   influence  of  on  the  form  of 
distribution,  52  f.  ;  57  f. 


Skewness,  31  f.,  38,  54  f.,  57  f.;  from 
mixture,  54  f.  ;  from  selection,  53 

Spearman,  C,  129 

Standard  deviation,  of  a  distribution, 
38,  59,  60,  76  ff.,  80,  142 ;  of  a  meas- 
ure, 139  ff. 

Subjectivity  of  measurements,  8  ff. 

Surface  of  frequency,  see  Distribution 

Technique  of  calculating  measures,  of 
change,  106  ff.  ;  of  central  tendency, 
71  ff.  ;  81  f.  ;  of  difference,  99  ff.,1  55 
f. ;  of  relationship,  110  ff. ;  of  relia- 
bility, 139  ff.,  153  f.  ;  of  transmuting 
measures  by  relative  position,  85  ff. ; 
of  use  of  tables  of  frequency,  147  ff.  ; 
of  variability,  75  ff. 

Transmutation,  of  measures  by  position, 
21,  85  ff.,  152  f. 

Units  of  measurement,  5,  7  ff.,  85  ff. 

Variability,  6,  22  ff.,  33  f.,  38,  59  f.  ; 
and  the  measurement  of  differences, 
98  ;  calculation  of  measures  of,  75  ff. ; 
causes  of,  61  ff. ;  comparison  of  groups 
in,  102  f. ;  of  a  divergence  as  a  measure 
of  reliability,  137  ff. ;  of  relationships, 
110  ff. ;  reliability  of  measures  of,  142 

Variable  errors,  157  ff. 

Weights,  161  f. 
Wilcox,  W.  F.  47 
Wood,  G.  H.,  24,  127. 

Yule,  G.  XL,  74,  206 

•Zero  points,  15  f.,  98,  114,  116 


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